Power functions

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Overview

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Exercises

Exercise 1

Given the function $f$ with $f (x) = 40 - \frac{100}{{(x + 10)}^{3}}$ .

Show that $f$ is a power function.

Determine the asymptotes of the graph of $f$ .

Write down the domain and range of $f$ .

Algebraically determine the x-intersects of the graph of $f$ .

Solve, rounding to two decimals: $f (x) \geq x$ .

Exercise 2

Algebraically solve the following inequalities.

$\frac{16}{x^{4}} \geq \frac{1}{2} x$

$\frac{2 x}{x - 10} + 20 < 100$

Exercise 3

Given the function $g$ with $g (x) = 20 x^{2} \sqrt{x} - 100$ .

Show that $g$ is a power function.

Write down the domain and range of $g$ .

Algebraically determine the x-intersect of the graph of $g$ .

Solve, rounding to two decimals: $g (x) \geq x$ .

Exercise 4

Algebraically solve the following inequalities.

$16 \sqrt[4]{x} \geq \frac{1}{2} x$

$2 \sqrt{2 x - 40} + 20 < 100$

Exercise 5

Given the functions $f$ and $g$ with $f (x) = \frac{10}{x \sqrt{x}} + 100$ and $g (x) = \frac{10 x}{\sqrt{x}}$ .

Both functions are power functions. Using the power of the functions, explain why the graph of $f$ is always decreasing and that of $g$ is increasing.

What are the asymptotes of these functions?

Solve, rounding to two decimals: $f (x) \geq g (x)$ .

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