Question

which equations represent nonlinear functions?

\\(k(x) = \frac{8}{x}\\)

\\(h(x) = \frac{x}{8}\\)

\\(g(x) = 8x\\)

\\(f(x) = 8x^2 + 8x\\)

\\(m(x) = 8^x\\)

\\(j(x) = 8\\)

Explanation:

⚡ Using what you learned: understanding linear functions

Step 1: Define linear and nonlinear functions

A linear function can be written in the form:

$$ y = mx + b $$

where the variable \( x \) has an exponent of exactly \( 1 \) and is not in the denominator, under a radical, or in an exponent.

Any function that cannot be written in this form is nonlinear.

Step 2: Analyze each option

• \( k(x) = \frac{8}{x} \): The variable \( x \) is in the denominator (which is \( 8x^{-1} \)). This is a rational function, so it is nonlinear.
• \( h(x) = \frac{x}{8} \): This can be rewritten as \( \frac{1}{8}x \), which fits the linear form \( mx + b \) with \( m = \frac{1}{8} \) and \( b = 0 \). This is linear.
• \( g(x) = 8x \): This fits the linear form with \( m = 8 \) and \( b = 0 \). This is linear.
• \( f(x) = 8x^2 + 8x \): The variable \( x \) has an exponent of \( 2 \). This is a quadratic function, so it is nonlinear.
• \( m(x) = 8^x \): The variable \( x \) is in the exponent. This is an exponential function, so it is nonlinear.
• \( j(x) = 8 \): This fits the linear form with \( m = 0 \) and \( b = 8 \) (a horizontal line). This is linear.

Answer:

The equations that represent nonlinear functions are:

• \( k(x) = \frac{8}{x} \)
• \( f(x) = 8x^2 + 8x \)
• \( m(x) = 8^x \)