Question

f(x) = log(|x + 4|) - 10
asymptote:
domain: range:
end behavior:

Explanation:

Step1: Find the Asymptote

The function is \( f(x)=\log(|x + 4|)-10 \). For the logarithmic function \( y = \log(|x - h|)+k \), the vertical asymptote is at \( x=h \) (where the argument of the log is zero). Here, the argument \( |x + 4|=0 \) when \( x=-4 \). So the vertical asymptote is \( x=-4 \).

Step2: Find the Domain

The domain of a logarithmic function \( \log(u) \) requires \( u>0 \). Here \( u = |x + 4| \). The absolute value \( |x + 4|>0 \) when \( x
eq - 4 \). So the domain is all real numbers except \( x=-4 \), which can be written in interval notation as \( (-\infty,-4)\cup(-4,\infty) \).

Step3: Find the Range

The range of \( \log(|x|) \) is all real numbers (\( (-\infty,\infty) \)) because as \( x \) approaches \( 0 \) from the right or left, \( \log(|x|) \) approaches \( -\infty \), and as \( |x| \) increases, \( \log(|x|) \) approaches \( \infty \). The transformation \( \log(|x + 4|)-10 \) is a vertical shift down by 10 units, but vertical shifts do not affect the range (they just shift the graph up or down). So the range of \( f(x) \) is all real numbers, \( (-\infty,\infty) \).

Step4: Find the End Behavior

• As \( x

ightarrow\infty \): \( |x + 4|\approx x \) (for large \( x \)), so \( f(x)=\log(x)-10 \). As \( x
ightarrow\infty \), \( \log(x)
ightarrow\infty \), so \( f(x)
ightarrow\infty - 10=\infty \).

• As \( x

ightarrow-\infty \): \( |x + 4|=-x - 4\approx -x \) (for large negative \( x \)), so \( f(x)=\log(-x)-10 \). As \( x
ightarrow-\infty \), \( -x
ightarrow\infty \), so \( \log(-x)
ightarrow\infty \), and \( f(x)
ightarrow\infty - 10=\infty \).

• As \( x

ightarrow - 4^{+} \) (approaching -4 from the right): \( |x + 4|=x + 4
ightarrow0^{+} \), so \( \log(x + 4)
ightarrow-\infty \), and \( f(x)=\log(x + 4)-10
ightarrow-\infty - 10=-\infty \).

• As \( x

ightarrow - 4^{-} \) (approaching -4 from the left): \( |x + 4|=-(x + 4)
ightarrow0^{+} \), so \( \log(-(x + 4))
ightarrow-\infty \), and \( f(x)=\log(-(x + 4))-10
ightarrow-\infty - 10=-\infty \).

Answer:

s:

• Asymptote: \( \boldsymbol{x=-4} \)
• Domain: \( \boldsymbol{(-\infty,-4)\cup(-4,\infty)} \)
• Range: \( \boldsymbol{(-\infty,\infty)} \)
• End Behavior: As \( \boldsymbol{x

ightarrow\pm\infty} \), \( \boldsymbol{f(x)
ightarrow\infty} \); as \( \boldsymbol{x
ightarrow - 4^{\pm}} \), \( \boldsymbol{f(x)
ightarrow-\infty} \)