Question

what are the features of the function $f(x) = \log_{2}(x + 2)$ graphed below? (image of the graph is shown with x-axis from -12 to 12 and y-axis from -12 to 12, the graph of the logarithmic function is present)
answer attempt 1 out of 2
the function $f(x)$ is dropdown function with a dropdown asymptote of dropdown $\square$. the range of the function is dropdown, and it is dropdown on its domain of dropdown. the end behavior on the left side is as dropdown, dropdown, and the end behavior on the right side is as dropdown, dropdown.

Explanation:

Step1: Determine the type of function

The function \( f(x)=\log_2(x + 2) \) is a logarithmic function. Logarithmic functions are one - to - one (since the logarithmic function \( y = \log_a(x) \) with \( a>0,a
eq1 \) is one - to - one) and increasing when the base \( a>1 \) (here \( a = 2>1 \)).

Step2: Find the vertical asymptote

For a logarithmic function \( y=\log_a(x - h)+k \), the vertical asymptote is \( x = h \). In the function \( f(x)=\log_2(x + 2)=\log_2(x-(- 2)) \), the vertical asymptote is \( x=-2 \).

Step3: Determine the range of the function

The range of a logarithmic function \( y = \log_a(x) \) is all real numbers (\( (-\infty,\infty) \)). For the function \( f(x)=\log_2(x + 2) \), the range is also all real numbers, i.e., \( (-\infty,\infty) \) or \( \mathbb{R} \).

Step4: Determine the monotonicity

Since the base \( a = 2>1 \), the function \( f(x)=\log_2(x + 2) \) is increasing on its domain.

Step5: Determine the domain of the function

For the logarithmic function \( \log_b(u) \), the argument \( u>0 \). So for \( f(x)=\log_2(x + 2) \), we need \( x + 2>0 \), which gives \( x>-2 \). So the domain is \( (-2,\infty) \).

Step6: Determine the end - behavior

• Left - hand end - behavior: As \( x\to - 2^+ \) (approaching - 2 from the right), \( x + 2\to0^+ \), and \( \log_2(x + 2)\to-\infty \).
• Right - hand end - behavior: As \( x\to\infty \), \( x + 2\to\infty \), and \( \log_2(x + 2)\to\infty \).

Answer:

The function \( f(x) \) is a \(\boldsymbol{\text{logarithmic}}\) function with a \(\boldsymbol{\text{vertical}}\) asymptote of \(\boldsymbol{x=-2}\). The range of the function is \(\boldsymbol{(-\infty,\infty)}\), and it is \(\boldsymbol{\text{increasing}}\) on its domain of \(\boldsymbol{(-2,\infty)}\). The end behavior on the LEFT side is as \(\boldsymbol{x\to - 2^+}\), \(\boldsymbol{f(x)\to-\infty}\), and the end behavior on the RIGHT side is as \(\boldsymbol{x\to\infty}\), \(\boldsymbol{f(x)\to\infty}\).