Definition of Inverse Functions
Definition of inverse functions is a Grade 7 advanced math concept from Yoshiwara Intermediate Algebra where students learn that two functions f and g are inverses if f(g(x)) = x and g(f(x)) = x. The inverse function reverses the input-output relationship of the original function.
Key Concepts
Property Two functions are inverse functions if each one undoes the effect of the other. The graphs of inverse functions are symmetric about the line $y = x$. If we interchange the variables in the function, we get an equivalent formula for its inverse. For example, $y = \sqrt[3]{x}$ if and only if $x = y^3$.
Examples The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$, taking the fifth power gives $2^5=32$. The fifth root of 32 is $\sqrt[5]{32}=2$, our original number.
The functions $f(x) = x+7$ and $g(x) = x 7$ are inverses. If you take a number, say 20, then $f(20) = 27$. Applying the inverse gives $g(27) = 20$, returning to the start.
Common Questions
What is an inverse function?
The inverse of f(x), written f^(-1)(x), reverses the input and output. If f(a) = b, then f^(-1)(b) = a.
How do you verify two functions are inverses?
Compute f(g(x)) and g(f(x)). If both equal x, then f and g are inverse functions of each other.
How do you find the inverse of a function algebraically?
Replace f(x) with y, swap x and y, then solve for y. The result is f^(-1)(x).
What does the graph of an inverse function look like?
The graph of f^(-1) is the reflection of the graph of f across the line y = x.