Question

the function f(x)=2x + 3 is one - to - one.
a. find an equation for f^(-1), the inverse function.
b. verify that your equation is correct by showing that f(f^(-1)(x))=x and f^(-1)(f(x))=x.
a. select the correct choice below and fill in the answer box(es) to complete your choice. (simplify your answer. use integers or fractions for any numbers in the expression.)
a. f^(-1)(x)=(x - 3)/2, for all x
b. f^(-1)(x)=, for x≥
c. f^(-1)(x)=, for x≠
d. f^(-1)(x)=, for x≤
b. verify that the equation is correct.
f(f^(-1)(x))=f() and f^(-1)(f(x))=f^(-1)() substitute.
= = simplify.

Explanation:

Step1: Find the inverse function

Let $y = f(x)=2x + 3$. Swap $x$ and $y$: $x = 2y+3$. Solve for $y$:
$x-3=2y$, so $y=\frac{x - 3}{2}$. Thus $f^{-1}(x)=\frac{x - 3}{2}$ for all $x$.

Step2: Verify $f(f^{-1}(x))=x$

Substitute $f^{-1}(x)$ into $f(x)$: $f(f^{-1}(x))=f(\frac{x - 3}{2})=2(\frac{x - 3}{2})+3$.
Simplify: $x-3 + 3=x$.

Step3: Verify $f^{-1}(f(x))=x$

Substitute $f(x)$ into $f^{-1}(x)$: $f^{-1}(f(x))=f^{-1}(2x + 3)=\frac{(2x + 3)-3}{2}$.
Simplify: $\frac{2x}{2}=x$.

Answer:

a. A. $f^{-1}(x)=\frac{x - 3}{2}$, for all $x$
b. $f(f^{-1}(x))=f(\frac{x - 3}{2})=2(\frac{x - 3}{2})+3=x$; $f^{-1}(f(x))=f^{-1}(2x + 3)=\frac{(2x + 3)-3}{2}=x$