Inverse of a Rational Function
Finding the inverse of a rational function and its domain
Problem: Find the inverse of the following function and its domain:\[
f(x) = \sqrt{x-10}
\]
Finding the inverse of a rational function and its domain
Finding the inverse of a rational function and its domain
Finding the inverse of a linear function and its domain
Restricting the domain of a function so it is one-to-one and has an inverse
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Finding the domains of functions
Finding the domains of expressions
Properties and derivatives of inverse trigonometric functions
Properties and derivatives of inverse trigonometric functions
Properties of inverse trig functions and the derivative of arctangent
Properties and derivatives of inverse trigonometric functions
Properties and derivatives of inverse trigonometric functions
Finding the domain of a function with a fraction and radicals
Finding the domain of a function with a quadratic inside a square root
Finding the composition of two functions with a square root and a fractional exponent
Explaining domain restrictions for denominators, even roots, and logarithms
Determining the domain and range of a relation and if the relation is a function
Determining the domain and range of a relation and if the relation is a function
Determining the domain and range of a relation and if the relation is a function
Finding the domain of a piecewise function and evaluating it
Finding the domain of a function that is a fraction containing exponentials and a square root
Using the Horizontal Line Test to determine if graphs represent invertible functions
Graphically verifying if two functions are inverses
Determining properties of an exponential function
Determining properties of an exponential function
Determining properties of an exponential function
Identifying the graph of an exponential function
Determining properties of an exponential function
Using a graph to evaluate a function
Review of limits and derivatives of inverse trigonometric functions
Review of limits and derivatives of inverse trigonometric functions
Review of limits and derivatives of inverse trigonometric functions
Using integrals to solve work problems and find the average value
Determining the domain of a three-dimensional vector function
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Determining if a graph represents a function and finding the domain and range
Finding the roots of a square root function
Finding the domain of a rational function
Finding the domain of a rational function
Finding the composition of two functions
Finding the composition of a rational function and a quadratic
Finding the composition of two functions with exponentials and logarithms
Finding the composition of two functions with a cube root and a fractional exponent
Finding the domain and x-intercepts of a logarithmic function
Finding the domain and x-intercepts of a logarithmic function
Solving a logarithmic equation using properties of logarithms
Solving a logarithmic equation using properties of logarithms
Finding the domain of even and odd roots
Finding the domain of a function with a denominator, square root, and logarithm
Solving an equation that contains rational expressions
Solving an equation that contains rational expressions
Explaining the general properties of even and odd root functions
Solving equations with multiple radicals
Explaining how to solve radical equations and then solving example problems
Solving equations with even roots and checking the solutions
Review of limits and derivatives of inverse trigonometric functions
Using a graph to find specific values of a function
Finding function values and expressions given the rule or equation of the function
Determining if functions are polynomials and finding their properties
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Determining the end behavior, real zeros, domain, and y-intercept for a polynomial
Finding the piecewise-defined function for a given graph along with its domain and range
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the properties of an exponential function including domain, range, end behavior, and intercepts
Finding the domain of a function that is a fraction containing an exponential and root
Finding the domain of a function that is a fraction containing an exponential and root
Evaluating a logarithm inside an exponential function
Finding the domain, range, end behavior, and intercepts for a logarithmic function
Finding the domain of a function containing a logarithm, exponential, radical, and fraction
Finding the domain of a function containing several logarithms
Finding the domain of a function containing a logarithm, exponential, and fraction
Finding the domain, intercepts, asymptotes, and holes of a rational function
Finding the domain, intercepts, asymptotes, and holes of a rational function
Finding the domain, intercepts, asymptotes, and holes of a rational function
Determining where a function with a fraction, square root, and logarithm is continuous
Determining where a piecewise function is continuous algebraically
Using the limit definition to find the derivative of a square root function
Finding the derivative of a function with a square root and exponential using the Product Rule
Finding the intervals where a function is increasing/decreasing and the local extrema
Finding intervals of increase/decrease and the local extema for revenue from a word problem
Finding the intervals of concavity and inflection points for a function with exponentials
Determining the intervals of concavity and inflection points from the graph of a function
Using the second derivative test to find local extrema for a function
Finding the absolute maximum and minimum of a rational function on two closed intervals
Finding the absolute maximum and absolute minimum of a rational function on two closed intervals
Finding the selling price of trucks to maximize profit given the cost and price-demand functions
Finding the dimensions of the box with the largest volume given the amount of material to make the box
Finding the minimum cost of a box given the volume and cost of materials
Using geometric formulas for circles, triangles, and rectangles to find the exact area under a curve
Using derivatives to find the velocity and acceleration functions from the position function
Simplifying an expression with radicals
Rationalizing the denominator of a fraction with square roots
Rationalizing the denominator of a fraction with square roots
Calculating the difference quotient for function with a square root
Determining if an equation represents a function
Evaluating a piecewise function
Identifying the parent function and transformations of a function
Identifying the parent function and transformations of a function
Determining the properties of a rational function and graphing it
Determining the properties of a rational function and graphing it
Finding combinations of functions and their domains
Determining the properties of a logarithmic function
Determining the graph of a logarithmic function
Determining the properties of a logarithmic function
Determining the graph of a logarithmic function
Solving an equation with an exponential function
Solving an equation with exponential functions by factoring
Determining if a function is even or odd
Cartesian equations and parametric equations of curves
Derivatives and vectors with some physics applications
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Continuity of Functions and the Intermediate Value Theorem
Approximation and Newton's Method, and limits and derivatives of exponential functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives and vectors with some physics applications
Derivatives of exponential and logarithmic functions and the exponential model
Derivatives and vectors with some physics applications
Derivatives of exponential and logarithmic functions and the exponential model
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Riemann sums, including approximating areas under curves.
Using Reimann sums and the Fundamental Theorem of Calculus
Using Riemann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Finding the derivative of a function using the limit definition
Using a linear approximation to approximate the value of a function
Review of work and average value
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Converting parametric equations into a Cartesian equation and graphing
Finding a Taylor Polynomial for a function
Finding the area of a region enclosed by two curves
Finding the area of a region bounded by curves
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Integrating a function with a square root using u-substitution
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Cartesian equations and parametric equations of curves
Derivatives and vectors with some physics applications
Reviewing the chain rule and the derivatives and limits of trigonometric functions
Related rates problems, differentials, linear and quadratic approximations
Proving a property of scalar multiplication for limits using the epsilon-delta definition and using the Squeeze Theorem for Limits.
Continuity of Functions and the Intermediate Value Theorem
Approximation and Newton's Method, and limits and derivatives of exponential functions
Using the limit definition to find derivatives of functions and vector functions
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives and vectors with some physics applications
Derivatives of exponential and logarithmic functions and the exponential model
Proving facts about logarithms and exponentials including the derivative of an exponential with an arbitrary base
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Riemann sums, including approximating areas under curves.
Using Reimann sums and the Fundamental Theorem of Calculus
The definition and properties of definite integrals
Using Riemann sums and the Fundamental Theorem of Calculus
Examples of integration by substitution
Review of work and average value
Reviewing u-Substitution
Review of finding the radius and interval of convergence for a power series
Review of Taylor and Maclaurin Series and their properties
Review of Taylor Polynomials and Taylor's Inequality
Finding a Taylor Polynomial for a function
Finding a vector equation for the tangent line to a three-dimensional vector function
Evaluating and then finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Sketching the level curves for a function of two variables
Sketching the level curves for a function of two variables
Explaining the Pythagorean Theorem and using it to find a missing side in a right triangle
Writing the domain of functions with radicals in interval notation
Solving a rational equation and checking the solutions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
Solving an equation with logarithmic functions
test
Finding the properties of a quadratic function such as the vertex, axis of symmetry, intercepts, etc.
Finding the domain, intercepts, holes, and vertical asymptotes for a rational function
Finding the domain of a piecewise-defined function and evaluating it for given x-values
Finding properties of a polynomial function such as the degree, end behavior, y-intercept, etc.
Finding properties of a polynomial function such as the degree, end behavior, y-intercept, etc.
Stating the domain in interval notation of a function with radicals, fractions, and exponentials
Find the domain of a function in interval notation with a fraction and logarithm
Writing the equation of a line passing through two points
Writing the equation of a line through two points in slope-intercept form
Writing the equation of a graphed line in standard form
Factoring a quadratic polynomial
Factoring a quadratic function
Factoring a cubic polynomial by grouping
Finding the greatest common factor of a polynomial and then factoring it out
Factoring a quadratic where the leading coefficient is 1
Factoring quadratics where the leading coefficient is not 1
Factoring several polynomials by grouping
Factoring a difference of two squares
Factoring a sum of cubes and a difference of cubes
Simplifying a rational expression
Adding and then simplifying two rational expressions
Multiplying two rational expressions and then simplifying
Dividing rational expressions and then simplifying the answer
Adding two rational expressions and then simplifying the answer
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Matching a rational function with its graph
Using the quadratic formula to find the roots of a function
Finding the roots of a cubic polynomial
Finding the roots of a function with an absolute value
Finding and simplifying the difference quotient for a quadratic function
Finding and simplifying the difference quotient for a rational function
Finding and simplifying the difference quotient for a square root function
Solving an inequality with a polynomial by factoring and using a sign diagram
Solving an inequality with a rational expression by factoring and using a sign diagram
Solving an inequality with two rational expressions using a sign diagram
Solving an inequality with a rational expression using a sign diagram
Solving an inequality with a quadratic by factoring and using a sign diagram
Finding the absolute maximum or absolute minimum of a quadratic
Writing the standard form of quadratic function given its vertex and one other point
Finding the component form and magnitude for a vector between two points
Performing calculations with vectors including addition, subtraction, and scalar multiplication
Calculating the dot product for three pairs of vectors
Finding a unit vector in the direction of a given vector
Solving a system of two linear equations using the method of substitution
Solving a system of two linear equations using the method of substitution
Solving a system of two linear equations
Solving a system on two equations where one is linear and one is quadratic
Solving a equation with an exponential function
Solving an equation with two exponential functions with different bases
Solving an equation with exponential functions by factoring
Solving a equation with exponential functions by factoring
Finding the size of a population of rabbits with exponential growth
Finding the size of a population of rabbits with exponential decay
Finding the horizontal asymptote of an exponential function
Finding the horizontal asymptote of an exponential function
Finding the horizontal asymptote of an exponential function
Expanding a logarithmic expression using logarithm properties
Condensing logarithmic expressions into a single logarithm
Finding the exact value of logarithms using logarithmic properties
Finding the vertical asymptote of a logarithmic function
Finding the vertical asymptote of a logarithmic function
Finding the values of all six trigonometric functions of an angle given a right triangle
Evaluating the other five trigonometric functions for an acute angle given the value of sine
Evaluating the other five trigonometric functions given sine and that tangent is negative
Finding the solutions to an equation with tangent in a given interval
Solving a trigonometric equation with sine and cosine on a given interval
Solving an quadratic-like equation with sine by factoring
Solving an equation with cosine on a given interval
Solving a trigonometric equation with sine on a given interval
Sketching one period of a transformed sine graph
Sketching one period of a cosine function with transformations
Finding the amplitude, period, phase shift, and vertical shift for a given sine function
Writing a cosine function for a given graph
Defining variables in Python and using the variables in equations
Finding the domain of functions with even and odd roots
Finding the domain of several rational functions
Finding the domain of a rational function
Explaining how to find any holes and vertical asymptotes of a rational function
Finding the domain of a rational function using factoring
Finding the domain of a logarithmic function
Finding the domain of a function with a natural logarithm and denominator
Finding the domain of a function with a square root in the denominator
Finding the domain of a function with a root and a logarithm in the denominator
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Explaining the graph and properties for the parent function of quadratics
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Discussing parent functions of rational functions and their properties
Explaining how to multiply and divide rational expressions
Performing algebraic operations with rational expressions and simplifying the answer
Solving equations that contain rational expressions
How to rewrite radical terms as power terms
Solving equations with odd radicals
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Explaining how to solve equations with power and exponential terms
Explaining the basic properties of logarithmic functions
Expanding logarithmic expressions using logarithm rules
Condensing a logarithmic expression into a single logarithm
Explaining how to solve logarithmic equations
Solving logarithmic equations with a single logarithm term
Solving logarithmic equations with multiple logarithm terms
Solving logarithmic equations with multiple logarithm terms on both sides
Evaluating a double integral over a given region by splitting it into two double integrals
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Explaining the basic properties of logarithmic functions