I still do not understand why the final answer is not

4-[-16,20] = [4+16, 4-20] = [20, -16]

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Step 1/2

Here's a step-by-step explanation of the math in bold:

1. Understanding the Function:

The function in question is q(x)=42n(5x).

It's a composition of two functions:

n(x), whose range is given as [8,10].

42x, which involves multiplication by 2 and subtraction by 4.

2. Analyzing the First Transformation:

Explanation:

The expression 2n(5x)first involves applying n(x)5x.

The range of n(5x) will still be [8,10], as the range of n(x) doesn't change when its input is shifted or scaled.

Step 2/2

3. Applying Multiplication:

The range is then multiplied by 2:[8,10]2=[16,20].

This doubles the width of the range, stretching it horizontally.

4. Performing Subtraction:

Finally, 4 is subtracted from the entire range:[16,20]4=[20,24].

This shifts the range vertically downward by 4 units.

5. Final Range:

Explanation:

Therefore, the range of q(x)is[20,24].

Final answer

In essence, the functionq(x) takes the output ofn(x), stretches it horizontally by a factor of 2, and then shifts it downward by 4 units. This process results in the final range of [20,24].