The function $$ f(x)=2 x+5 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x) ight)=x $$ of 3 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. $$ f^{-1}(x)=\square $$, for $$ x
eq \square $$ B. $$ f^{-1}(x)=\square $$, for all $$ x $$ C. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ D. $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$

Question

The function $$ f(x)=2 x+5 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)ight)=x $$ of 3 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. $$ f^{-1}(x)=\square $$, for $$ x 
eq \square $$ B. $$ f^{-1}(x)=\square $$, for all $$ x $$ C. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ D. $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$

UpStudy AI · Accepted Answer

1. To find $$ f^{-1}(x) $$ for $$ f(x)=2x+5 $$, start by letting
   $$
   y = 2x + 5.
   $$
2. Solve for $$ x $$ in terms of $$ y $$:
   $$
   y - 5 = 2x \implies x = \frac{y-5}{2}.
   $$
3. Swap $$ x $$ and $$ y $$ to obtain the inverse function:
   $$
   f^{-1}(x) = \frac{x-5}{2}.
   $$

4. Verify the inverse by showing $$ f(f^{-1}(x)) = x $$:
   $$
   f(f^{-1}(x)) = f\left(\frac{x-5}{2}\right) = 2\left(\frac{x-5}{2}\right) + 5 = (x-5) + 5 = x.
   $$

5. Since the inverse function holds for all $$ x $$, the correct choice is:

**B.** $$ f^{-1}(x)=\frac{x-5}{2} $$, for all $$ x $$.
$$ f^{-1}(x) = \frac{x - 5}{2} $$

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