Exponential and Logarithmic Functions
5.1 Inverse Functions
Undoing a Function
Inverse functions reverse the original function process.
Definition: Inverse Function
Functions $f$ and $g$ are inverses if
$$f(g(x))=x\quad\text{and}\quad g(f(x))=x$$
on appropriate domains.
Fact: Horizontal Line Test
A function has an inverse function if it is one-to-one, which graphically means every horizontal line crosses at most once.
Example: Find an inverse algebraically
Find $f^{-1}(x)$ for
$$f(x)=3x-5.$$
Show Solution
Set $y=3x-5$.
Swap variables:
$$x=3y-5.$$
Solve for $y$:
$$x+5=3y\Rightarrow y=\frac{x+5}{3}. $$
So
$$f^{-1}(x)=\frac{x+5}{3}. $$
MyOpenMath: Practice: finding inverse functions (placeholder)
Verifying Inverses with Composition
Composition confirms whether two formulas truly undo each other.
Example: Verify inverses
Let
$$f(x)=2x+1,\qquad g(x)=\frac{x-1}{2}. $$
Verify they are inverses.
Show Solution
Compute both compositions:
$$f(g(x))=2\left(\frac{x-1}{2}\right)+1=x-1+1=x,$$ $$g(f(x))=\frac{(2x+1)-1}{2}=\frac{2x}{2}=x.$$
Both simplify to $x$, so the functions are inverses.