Question

Determine whether f has an inverse function. If it does, find the inverse function and state any restrictions on its domain.

f(x) = x - 7 (over)
x +4
a. No inverse function
b. Inverse function: f^(-1)(x) = (x + 7)/(x - 4), domain restriction: x ≠ 4
c. Inverse function: f^(-1)(x) = (x - 7)/(x + 4), domain restriction: x ≠ -4
d. No inverse function

Answer 1

Yes, the function f(x) = (x - 7)/(x + 4) does have an inverse function. The inverse function is f^(-1)(x) = (x + 7)/(x - 4) with a domain restriction of x ≠ 4.

Step-by-step explanation:

To determine whether the function f(x) = \frac{x - 7}{x + 4} has an inverse, we need to check if the function is one-to-one. A one-to-one function passes the Horizontal Line Test, which means that any horizontal line would intersect the graph of the function at at most one point. Since this is a rational function and not a constant, it will pass the Horizontal Line Test. Therefore, f(x) does have an inverse.

To find the inverse function f-1(x), we swap x and y and solve for y:

• Let y = f(x).
• Therefore, x = \frac{y - 7}{y + 4}.
• Multiply both sides by y + 4 to get x(y + 4) = y - 7.
• Distribute x to get xy + 4x = y - 7.
• Rearrange to get xy - y = -4x - 7.
• Factor out y to get y(x - 1) = -4x - 7.
• Divide by x - 1 to solve for y, getting y = \frac{-4x - 7}{x - 1}.

After simplifying, we obtain f-1(x) = \frac{x + 7}{x - 4}. However, because the original function had a vertical asymptote at x = -4, the inverse function will have a domain restriction of x ≠ 4 to avoid division by zero.