Question

what are the features of the function $f(x) = 3^x - 4$ graphed below?

graph of the function

answer attempt 1 out of 2
the function $f(x)$ is ____ function with a __ asymptote of __. the range of the function is __, and it is __ on its domain of __. the end behavior on the left side is as __, __, and the end behavior on the right side is as __, ____.

Explanation:

Step1: Identify the function type

The function \( f(x) = 3^x - 4 \) is an exponential function. Exponential functions have the form \( a^x + k \), where \( a>0,a
eq1 \). Here, \( a = 3 \) (which is \(>1\)) and \( k=-4 \).

Step2: Determine the asymptote

For an exponential function \( y = a^x + k \), the horizontal asymptote is \( y = k \). For \( f(x)=3^x - 4 \), the horizontal asymptote is \( y=-4 \).

Step3: Find the domain

The domain of an exponential function \( a^x + k \) is all real numbers, \( (-\infty,\infty) \), because we can plug in any real number for \( x \) into \( 3^x \).

Step4: Determine the range

For \( y = 3^x \), the range is \( (0,\infty) \). When we subtract 4, we shift the graph down 4 units. So the range of \( f(x)=3^x - 4 \) is \( (-4,\infty) \).

Step5: Determine if the function is increasing or decreasing

Since the base \( a = 3>1 \), the exponential function \( 3^x \) is increasing. Subtracting 4 (a vertical shift) does not change the increasing/decreasing nature, so \( f(x)=3^x - 4 \) is increasing on its domain.

Step6: Analyze end - behavior

• Left - hand end - behavior: As \( x

ightarrow-\infty \), \( 3^x
ightarrow0 \) (because for \( a > 1 \), as \( x
ightarrow-\infty \), \( a^x
ightarrow0 \)). Then \( f(x)=3^x - 4
ightarrow0 - 4=-4 \). So as \( x
ightarrow-\infty \), \( f(x)
ightarrow - 4 \).

• Right - hand end - behavior: As \( x

ightarrow\infty \), \( 3^x
ightarrow\infty \) (because for \( a>1 \), as \( x
ightarrow\infty \), \( a^x
ightarrow\infty \)). Then \( f(x)=3^x - 4
ightarrow\infty-4=\infty \). So as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \).

Answer:

The function \( f(x) \) is an exponential function with a horizontal asymptote of \( y = - 4 \). The range of the function is \( (-4,\infty) \), and it is increasing on its domain of \( (-\infty,\infty) \). The end behavior on the LEFT side is as \( x
ightarrow-\infty \), \( f(x)
ightarrow - 4 \), and the end behavior on the RIGHT side is as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \).