Answered

Drag each tile to the correct location on the image. Each tile can be used more than once, but not all tiles will be used.

Consider the function [tex]f(x) = \sqrt[3]{8x} + 4[/tex].

To determine the inverse of function [tex]f[/tex], change [tex]f(x)[/tex] to [tex]y[/tex], switch [tex]x[/tex] and [tex]y[/tex], and solve for [tex]y[/tex].

The resulting function can be written as
[tex]
f^{-1}(x) = (x - \quad)^3
[/tex].



Answer :

GinnyAnswer
To determine the inverse of the function [tex]\( f(x) = \sqrt[2]{8x} + 4 \)[/tex], we need to go through the following detailed steps:

1. Rewrite [tex]\( f(x) \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[2]{8x} + 4 \][/tex]

2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[ x = \sqrt[2]{8y} + 4 \][/tex]

3. Isolate the square root term:
[tex]\[ x - 4 = \sqrt[2]{8y} \][/tex]

4. Square both sides to get rid of the square root:
[tex]\[ (x - 4)^2 = 8y \][/tex]

5. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{(x - 4)^2}{8} \][/tex]

Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] in terms of [tex]\( y \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^2}{8} \][/tex]

To match the form given in the problem [tex]\( f^{-1}(x) = (x - \quad)^3 \)[/tex], we need to identify what [tex]\( (x - 4)^2 / 8 \)[/tex] would be equivalent to if written in another form.

Given the structure of the problem, the blank [tex]\((x - \quad)^3\)[/tex] should correspond to shifting the value used in the transformation. The inverse function after solving shows the shift was by the value 4.

Therefore:
[tex]\[ f^{-1}(x)=(x-4)^3 \][/tex]

The missing value in the blank is [tex]\( 4 \)[/tex].