Question

Valeria is given the function f(x)=3x/2+4. She makes these claims. 1. The inverse function if f-1(x)=2(x-4)/3 2. f(x)and f-1(x) intersect at (-8,-8) 3. The combined graphs of f(x) and f-1(x) form an image that is symmetric about the line y=x. Which of Valeria's statements is/are true?

Answer 1

The inverse of a function is given by the reflection of the function across the line y = x

Valeria's statements which are true are;

All three statements 1, 2, and 3 are true

The reason the all three statements are true is as follows:

The given function is presented as follows:

$f(x) = \dfrac{3 \cdot x}{2} + 4$

The inverse of the function is found as follows;

$y = \dfrac{3 \cdot x}{2} + 4$

2 × (y - 4) = 3·x

$x = \dfrac{2 \cdot (y - 4)}{3}$

Therefore, the inverse of the function, f⁻¹(x) is presented as follows;

$\mathbf{f^{-1} (x)} = \dfrac{2 \cdot (x - 4)}{3}$

Therefore, statement 1 is true

Statement 2. f(x) and f⁻¹(x) intersect at (-8, -8)

At (-8, -8), we have;

$\mathbf{f(x)} = \dfrac{3 \times (-8)}{2} + 4 = \mathbf{ -8}$

$\mathbf{f^{-1} (x)} = \dfrac{2 \times ((-8) - 4)}{3} = \dfrac{2 \times (-12)}{3} = \mathbf{-8}$

Therefore at the point (-8, -8), f(x) = f⁻¹(x) and they therefore intersect

Statement 2 is therefore true

Statement 3. The combined graph of f(x) and f⁻¹(x) form an image that is symmetric about the the line y = x

The above statement 3 is true because the graph of the inverse of a function is the reflection of the function about the line y = x

Therefore, all the statements 1, 2, and 3 are true

Learn more about the inverse of functions here:

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