Question

Vincent wrote an example of a proportion on the board: "If f(x) and y(x) are inverse functions of each other and f(x) = 2x+5, what is y(8)?"

A) -1
B) 3
C) 41
D) 8
E) 23

Answer 1

If f(x) and y(x) are inverse functions of each other and f(x) = 2x+5, what is y(8) is 3 (option B)

Step-by-step explanation:

If \(f(x)\) and \(y(x)\) are inverse functions, then \(f(x) = 2x + 5\) should be the inverse of \(y(x)\). To find \(y(8)\), substitute \(8\) into the equation \(f(x) = 2x + 5\) to find the value of \(y(8)\). So, \(y(8)\) would be equal to \(8 - 5\), resulting in \(y(8) = 3\). Therefore, the correct answer is option B.

Inverse functions are two functions that, when composed together, yield the identity function. In this case, since \(f(x)\) and \(y(x)\) are inverses, \(f(x) = 2x + 5\) and \(y(8) = 3\) satisfies this inverse relationship.

Understanding inverse functions is crucial in mathematics, especially in algebra and calculus, as they represent functions that "undo" each other's operations when composed together.

Answer 2

There appears to be an error in the question or answer choices, and C is the closest option to the calculated value of
`\( y(8) \).` Thus the correct option is c.

Step-by-step explanation:

Given that f(x) = 2x + 5 is an inverse function of y(x), we can find y(8) by substituting 8 for x in the expression for f(x).

`\[ f(x) = 2x + 5 \]`

`\[ f(8) = 2(8) + 5 = 16 + 5 = 21 \]`

Therefore, y(8) is 21. However, the answer choices provided do not include 21. To determine the correct answer, we need to understand the relationship between f(x) and y(x) in inverse functions.

If f(x) and y(x) are inverse functions, swapping x and y in the expression for f(x) gives the expression for y(x). In this case, we interchange x and y in the expression
`\( f(x) = 2x + 5 \):`

`\[ x = 2y + 5 \]`

Now, solve for y:

`\[ 2y = x - 5 \]`

`\[ y = (x - 5)/(2) \]`

Substitute x = 8 into this expression:

`\[ y(8) = (8 - 5)/(2) = (3)/(2) = 1.5 \]`

However, the provided answer choices do not include 1.5. The closest option is C) 41. It seems there might be an error in the question or the answer choices.