Question

A function f has the following verbal description: multiply by 3, add 5, and

Answer 1

Regarding the functions in this problem, it is found that:

a) The verbal description of the inverse is: Take the third root, subtract by 5 and divide by 3.

b) The formulas are:

• f(x) = (3x + 5)³.

• $f^{-1}(x) = \frac{\sqrt[3]{x} - 5}{3}$

c) The compositions result in x, hence they are inverses.

How to obtain the inverse function?

The original function is described as follows:

• Multiply by 3: 3x.

• Add 5: 3x + 5.

• Take the third power: (3x + 5)³.

Hence the function is:

y = (3x + 5)³.

To obtain the inverse functions, we exchange x and y and then isolate y, hence:

x = (3y + 5)³

$\sqrt[3]{x} = \sqrt[3]{(3y+5)^3}$

$3y + 5 = \sqrt[3]{x}$

$3y = \sqrt[3]{x} - 5$

$y = \frac{\sqrt[3]{x} - 5}{3}$

$f^{-1}(x) = \frac{\sqrt[3]{x} - 5}{3}$

The verbal description of the inverse is:

• Take the third root. sqrt[3](x).

• Subtract by 5: sqrt[3](x) - 5.

• Divide by 3.

The compositions are given as follows:

• $f(f^{-1}(x)) = f\left(\frac{\sqrt[3]{x} - 5}{3}\right) = \left(3\left(\frac{\sqrt[3]{x} - 5}{3}\right + 5)\right)^3 = (\sqrt[3]{x})^3 = x$

• $f^{-1}(f(x)) = f^{-1}((3x + 5)^3) = \frac{\sqrt[3]{(3x + 5)^3} - 5}{3} = \frac{3x + 5 - 5}{3} = x$

More can be learned about inverse functions at brainly.com/question/11735394

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