Question

select the correct answer. which statement describes the end behavior of this function? r(x) = log(x - 2) a. as the value of x decreases, the value of r(x) moves toward positive infinity. b. as the value of x increases, the value of r(x) moves toward positive infinity. c. as the value of x increases, the value of r(x) moves toward negative infinity. d. as the value of x decreases, the value of r(x) moves toward a constant.

Explanation:

Step1: Recall properties of logarithmic functions

The function \( r(x) = \log(x - 2) \) is a logarithmic function with base 10 (since it's not specified, we assume base 10). The domain of \( \log(x - 2) \) is \( x - 2>0 \) or \( x > 2 \). The general form of a logarithmic function \( y=\log_b(x - h)+k \) (here \( h = 2,k = 0,b = 10\)) has the following end - behavior:

• For a logarithmic function \( y=\log_b(x - h) \) with \( b>1 \) (which is true for \( b = 10\)), as \( x

ightarrow\infty \), we know that the logarithmic function \( y = \log_b(u) \) where \( u=x - h \) and as \( x
ightarrow\infty \), \( u=x - h
ightarrow\infty \). Since the logarithmic function \( y=\log_b(u) \) with \( b > 1\) is an increasing function (its derivative \( y^\prime=\frac{1}{u\ln b}>0 \) for \( u>0 \)), as \( u
ightarrow\infty \), \( y=\log_b(u)
ightarrow\infty \).

• As \( x \) approaches 2 from the right (i.e., \( x

ightarrow2^+ \)), \( u=x - 2
ightarrow0^+ \), and for \( y = \log_{10}(u) \), as \( u
ightarrow0^+ \), \( y=\log_{10}(u)
ightarrow-\infty \).

Step2: Analyze each option

• Option A: As \( x \) decreases, but the domain of \( r(x)=\log(x - 2) \) is \( x>2 \). So \( x \) cannot decrease beyond 2 (in the domain). So this option is incorrect.
• Option B: As \( x \) increases, \( u=x - 2 \) increases. Since \( y = \log_{10}(u) \) is an increasing function (because the base \( 10>1 \)), as \( u

ightarrow\infty \) (when \( x
ightarrow\infty \)), \( \log_{10}(u)
ightarrow\infty \). So as \( x \) increases, \( r(x)=\log(x - 2) \) moves toward positive infinity. This option is correct.

• Option C: As \( x \) increases, since the function is increasing (base \( 10>1 \)), \( r(x) \) should move toward positive infinity, not negative infinity. So this option is incorrect.
• Option D: As \( x \) decreases (but in the domain \( x>2 \), \( x \) can only decrease towards 2), as \( x

ightarrow2^+ \), \( r(x)=\log(x - 2)
ightarrow-\infty \), not a constant. So this option is incorrect.

Answer:

B. As the value of \( x \) increases, the value of \( r(x) \) moves toward positive infinity.