Question

Find the correct answer in each box. Use numerals instead of words. if necessary, use / for tion bar(s). Find the inverse of the given function.

f(x)=5x+1/-x+7, x≠7

Answer 1

To find the inverse of the function f(x) = (5x + 1) / (-x + 7), switch x and y in the equation, solve for y, then check the derived inverse function to ensure correctness.

Step-by-step explanation:

To find the inverse of the given function f(x) = (5x + 1) / (-x + 7), where x ≠ 7, follow these steps:

1. First, replace f(x) with y: y = (5x + 1) / (-x + 7).
2. Next, switch the roles of x and y: x = (5y + 1) / (-y + 7).
3. Solve this equation for y to get the inverse function:
4. Multiply both sides by (-y + 7) to get rid of the fraction: x(-y + 7) = 5y + 1.
5. Distribute the x: -xy + 7x = 5y + 1.
6. Get all terms with y on one side: -xy - 5y = 1 - 7x.
7. Factor out the y: y(-x - 5) = 1 - 7x.
8. Finally, divide by (-x - 5) to isolate y: y = (1 - 7x) / (-x - 5).
9. Check the answer to ensure it makes sense by substituting values or verifying that it is the correct mathematical inverse.

The inverse function is f^{-1}(x) = (1 - 7x) / (-x - 5).