Questions: HW 2.7 Question 3, 2.7.15 Part 2 of 3 Points: 0 of 1 The function f(x)=3x+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(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)= for x ≠ B. f^(-1)(x)=(x-5)/3, for all 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.

Solution Steps

To find the inverse of the function \( f(x) = 3x + 5 \), we need to solve the equation \( y = 3x + 5 \) for \( x \). This involves isolating \( x \) on one side of the equation. Once we have the inverse function, we can verify it by checking that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

To find the inverse of the function \( f(x) = 3x + 5 \), we start by setting \( y = 3x + 5 \). Rearranging this equation to solve for \( x \) gives us:

\[ y - 5 = 3x \]

Dividing both sides by 3, we find:

\[ x = \frac{y - 5}{3} \]

Thus, the inverse function is:

\[ f^{-1}(x) = \frac{x - 5}{3} \]

To verify that our inverse function is correct, we need to check two conditions:

1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

Verification of the first condition:

Substituting \( f^{-1}(x) \) into \( f(x) \):

\[ f(f^{-1}(x)) = f\left(\frac{x - 5}{3}\right) = 3\left(\frac{x - 5}{3}\right) + 5 = x - 5 + 5 = x \]

Verification of the second condition:

Substituting \( f(x) \) into \( f^{-1}(x) \):

\[ f^{-1}(f(x)) = f^{-1}(3x + 5) = \frac{(3x + 5) - 5}{3} = \frac{3x}{3} = x \]

Both conditions are satisfied, confirming that our inverse function is correct.

Final Answer