Question

select the correct answer.

function (h) is nonlinear, and (h(4) = 2). which equation could represent function (h)?

(h(x) = 2^x - 14)
(h(x) = x^3 - 4)
(h(x) = \frac{1}{2}x + 3)
(h(x) = 2)

Explanation:

⚡ Using: understanding linear functions · 🆕 New: Evaluating Nonlinear Functions
Evaluating functions and identifying linear vs. nonlinear relationships.

Step 1: Understand the conditions

We are looking for a function \( h(x) \) that satisfies two conditions:

1. The function must be nonlinear (its graph is not a straight line; the variable \( x \) is not raised to the power of 1 in a simple linear form \( mx + b \)).
2. The function must satisfy \( h(4) = 2 \) (when we substitute \( x = 4 \), the output must be \( 2 \)).

Step 2: Identify and eliminate linear functions

Let's look at the options to see which ones are linear:

• \( h(x) = \frac{1}{2}x + 3 \) is a linear function because it is in the slope-intercept form \( y = mx + b \).
• \( h(x) = 2 \) is a constant function, which is also a type of linear function (a horizontal line).

This leaves us with two nonlinear options to test:

• \( h(x) = 2^x - 14 \) (exponential)
• \( h(x) = x^3 - 4 \) (cubic)

Step 3: Test the remaining options with \( x = 4 \)

Let's evaluate \( h(4) \) for both nonlinear options:

Option 1: \( h(x) = 2^x - 14 \)

$$ h(4) = 2^4 - 14 $$

$$ h(4) = 16 - 14 $$

$$ h(4) = 2 $$

This matches our condition \( h(4) = 2 \).

Option 2: \( h(x) = x^3 - 4 \)

$$ h(4) = 4^3 - 4 $$

$$ h(4) = 64 - 4 $$

$$ h(4) = 60 $$

This does not equal \( 2 \).

Answer:

\( h(x) = 2^x - 14 \)