The function $$ f(x)=5 x+7 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x) ight)=x $$ and $$ f^{-1}(f(x))=x $$. a. Select the correct choice below and fill in the answer box (es) to complete your choi (Simplify your answer. Use integers or fractions for any numbers in the expression.) $$ f^{-1}(x)= $$ for $$ x \leq $$ A. $$ f^{-1}(x)=\frac{x-7}{5} $$, for all $$ x $$ $$ f^{-1}(x)= $$ $$ f^{-1}(x)= $$ for $$ x \geq $$ for $$ x
eq $$ b. Verify that the equation is correct. $$ \left.f^{-1}(x) ight)=f(\square) $$

Question

The function $$ f(x)=5 x+7 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)ight)=x $$ and $$ f^{-1}(f(x))=x $$. a. Select the correct choice below and fill in the answer box (es) to complete your choi (Simplify your answer. Use integers or fractions for any numbers in the expression.) $$ f^{-1}(x)= $$ for $$ x \leq $$ A. $$ f^{-1}(x)=\frac{x-7}{5} $$, for all $$ x $$ $$ f^{-1}(x)= $$ $$ f^{-1}(x)= $$ for $$ x \geq $$ for $$ x 
eq $$ b. Verify that the equation is correct. $$ \left.f^{-1}(x)ight)=f(\square) $$

UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined:

### Part a: Find the Inverse Function

1. **Start with the function**:
   $$
   f(x) = 5x + 7
   $$

2. **Replace $$ f(x) $$ with $$ y $$**:
   $$
   y = 5x + 7
   $$

3. **Solve for $$ x $$** in terms of $$ y $$:
   - Subtract 7 from both sides:
     $$
     y - 7 = 5x
     $$
   - Divide by 5:
     $$
     x = \frac{y - 7}{5}
     $$

4. **Replace $$ y $$ with $$ x $$** to find the inverse function:
   $$
   f^{-1}(x) = \frac{x - 7}{5}
   $$

Thus, the inverse function is:
$$
f^{-1}(x) = \frac{x - 7}{5} \quad \text{for all } x
$$

### Part b: Verify the Inverse Function

We need to verify that:
1. $$ f(f^{-1}(x)) = x $$
2. $$ f^{-1}(f(x)) = x $$

#### Verification 1: $$ f(f^{-1}(x)) $$

1. Substitute $$ f^{-1}(x) $$ into $$ f(x) $$:
   $$
   f(f^{-1}(x)) = f\left(\frac{x - 7}{5}\right)
   $$
2. Calculate $$ f\left(\frac{x - 7}{5}\right) $$:
   $$
   = 5\left(\frac{x - 7}{5}\right) + 7
   $$
   $$
   = (x - 7) + 7
   $$
   $$
   = x
   $$

#### Verification 2: $$ f^{-1}(f(x)) $$

1. Substitute $$ f(x) $$ into $$ f^{-1}(x) $$:
   $$
   f^{-1}(f(x)) = f^{-1}(5x + 7)
   $$
2. Calculate $$ f^{-1}(5x + 7) $$:
   $$
   = \frac{(5x + 7) - 7}{5}
   $$
   $$
   = \frac{5x}{5}
   $$
   $$
   = x
   $$

Both verifications confirm that the inverse function is correct.

### Final Answers

a. The inverse function is:
$$
f^{-1}(x) = \frac{x - 7}{5} \quad \text{for all } x
$$

b. Verification:
$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
$$
a. The inverse function is $$ f^{-1}(x) = \frac{x - 7}{5} $$ for all $$ x $$. b. Verification: $$ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x $$

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