Inverse functions flip the input and output of original functions, undoing their effects. They're crucial for solving equations and modeling real-world situations where you need to reverse a process. Understanding inverses helps you grasp function behavior and relationships.
involve roots and have unique properties due to domain restrictions. Their inverses often involve exponents, making them useful in various applications. Mastering radical functions and their inverses enhances problem-solving skills in math and science.
Inverse Functions
Domain restriction for inverse functions
"undoes" the original function, denoted as f−1(x)
To find inverse: replace f(x) with y, swap x and y, solve for y, replace y with f−1(x)
Function must be one-to-one (injective) to have an inverse
Each codomain element paired with at most one domain element
Horizontal line test: if horizontal line intersects graph more than once, not one-to-one
Restricting domain can make function one-to-one
f(x)=x2 inverse not a function, fails horizontal line test
Restricting domain to x≥0, inverse becomes f−1(x)=x
Range restriction may also be necessary to ensure invertibility
Graphing and relationships of inverses
Inverse function graph is reflection of original across line y=x
If (a,b) on f(x) graph, then (b,a) on f−1(x) graph
Inverse function properties:
(f−1∘f)(x)=x and (f∘f−1)(x)=x, ∘ denotes function composition
f(x) domain is f−1(x) range, f(x) range is f−1(x) domain
Inverse functions have same symmetry as original with respect to y=x
Even function f(x) symmetric about y-axis, f−1(x) symmetric about y=x
Odd function f(x) symmetric about origin, f−1(x) also symmetric about origin
Types of Functions and Invertibility
Bijective functions (both injective and surjective) are
Surjective functions map onto the entire codomain
Invertible functions have a unique inverse that "undoes" the original function
Radical Functions and Their Inverses
Applications of radical function inverses
Radical functions involve variable under root (f(x)=x, g(x)=32x−1)
Radical function domain limited to values resulting in non-negative under radical
f(x)=x domain is x≥0
Radical function inverse found by steps for finding function inverse
f(x)=x inverse is f−1(x)=x2, domain restricted to x≥0
Application problems may include:
Solving for input given output value
Determining function and inverse domain and range
Graphing function and its inverse
Interpreting function and inverse meaning in problem context
Key Terms to Review (7)
Cubic functions: Cubic functions are polynomial functions of degree three, typically written in the form $f(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$. These functions can have up to three real roots and exhibit various shapes including an S-curve.
Inverse function: An inverse function reverses the effect of the original function, denoted as $f^{-1}(x)$. For a function $f(x)$ to have an inverse, it must be bijective (both injective and surjective).
Inverse of a radical function: The inverse of a radical function is found by swapping the dependent and independent variables and solving for the new dependent variable. It effectively undoes the operation of the original radical function.
Inverse of a rational function: The inverse of a rational function is a function that reverses the effect of the original rational function. If $f(x)$ is a rational function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Invertible functions: An invertible function is a function that has an inverse, meaning there exists another function that 'undoes' the effect of the original function. For a function to be invertible, it must be both one-to-one (injective) and onto (surjective).
Radical functions: A radical function is a function that involves a root, typically the square root, cube root, or higher-order roots of a variable. These functions are expressed in the form $f(x) = \sqrt[n]{x}$ where $n$ is the index of the root.
Surface area: Surface area is the total area that the surface of a three-dimensional object occupies. It is measured in square units and helps understand the extent of an object's exterior.