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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:
[tex]f(x) = \dfrac{3 \cdot x}{2} + 4[/tex]
The inverse of the function is found as follows;
[tex]y = \dfrac{3 \cdot x}{2} + 4[/tex]
2 × (y - 4) = 3·x
[tex]x = \dfrac{2 \cdot (y - 4)}{3}[/tex]
Therefore, the inverse of the function, f⁻¹(x) is presented as follows;
[tex]\mathbf{f^{-1} (x)} = \dfrac{2 \cdot (x - 4)}{3}[/tex]
Therefore, statement 1 is true
Statement 2. f(x) and f⁻¹(x) intersect at (-8, -8)
At (-8, -8), we have;
[tex]\mathbf{f(x)} = \dfrac{3 \times (-8)}{2} + 4 = \mathbf{ -8}[/tex]
[tex]\mathbf{f^{-1} (x)} = \dfrac{2 \times ((-8) - 4)}{3} = \dfrac{2 \times (-12)}{3} = \mathbf{-8}[/tex]
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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