Question

select the correct answer.
which statement describes the end behavior of the function $f(x) = 8|x - 1| + 13$?

a. as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

b. as $x$ approaches negative infinity, $f(x)$ approaches positive infinity.

c. as $x$ approaches positive infinity, $f(x)$ approaches negative infinity.

d. as $x$ approaches positive infinity, $f(x)$ is no longer continuous.

Explanation:

Step1: Recall absolute value function properties

The general form of an absolute value function is \( f(x) = a|x - h| + k \). For \( a>0 \), the graph opens upwards (like a "V" shape). Here, \( a = 8>0 \), so the function \( f(x)=8|x - 1| + 13 \) opens upwards.

Step2: Analyze end behavior for \( x\to-\infty \)

For \( x\to-\infty \), \( |x - 1|=-(x - 1)=-x + 1 \) (since \( x \) is very negative, \( x - 1 \) is negative, so absolute value makes it positive). Then \( f(x)=8(-x + 1)+13=-8x + 8 + 13=-8x + 21 \). As \( x\to-\infty \), \( -8x\to+\infty \) (because multiplying a large negative number by -8 gives a large positive number), so \( f(x)\to+\infty \).

Step3: Analyze end behavior for \( x\to+\infty \)

For \( x\to+\infty \), \( |x - 1|=x - 1 \) (since \( x \) is very positive, \( x - 1 \) is positive). Then \( f(x)=8(x - 1)+13=8x - 8 + 13=8x + 5 \). As \( x\to+\infty \), \( 8x\to+\infty \), so \( f(x)\to+\infty \).

Step4: Evaluate options

• Option A: Incorrect, as we saw \( f(x)\to+\infty \) when \( x\to-\infty \), not negative infinity.
• Option B: Correct, as \( x\to-\infty \), \( f(x)\to+\infty \).
• Option C: Incorrect, \( f(x)\to+\infty \) as \( x\to+\infty \).
• Option D: Incorrect, absolute value functions are continuous everywhere, so as \( x\to+\infty \), \( f(x) \) is still continuous.

Answer:

B. As x approaches negative infinity, f(x) approaches positive infinity.