Question

51 what is the domain and range of $f(x) = -3^x - 7$?
a domain $(-infty, infty)$
range $(-infty, infty)$
b domain $(-infty, infty)$
range $(-3, infty)$
c domain $(-infty, infty)$
range $(-7, infty)$
d domain $(-infty, infty)$
range $(-infty, -7)$

Explanation:

Step1: Analyze the Domain of \( f(x) = -3^x - 7 \)

The function \( y = 3^x \) is an exponential function, and the domain of any exponential function of the form \( a^x \) (where \( a>0, a
eq1 \)) is all real numbers, i.e., \( (-\infty, \infty) \). For the function \( f(x)=-3^x - 7 \), there are no restrictions on the input \( x \) (no division by zero, no square roots of negative numbers, etc.), so the domain of \( f(x) \) is also \( (-\infty, \infty) \).

Step2: Analyze the Range of \( f(x) = -3^x - 7 \)

First, recall the range of the parent exponential function \( y = 3^x \). For \( y = 3^x \), since \( 3>1 \), as \( x \to -\infty \), \( 3^x \to 0 \), and as \( x \to \infty \), \( 3^x \to \infty \). So the range of \( y = 3^x \) is \( (0, \infty) \).

Now, consider the transformation \( y=-3^x \). Multiplying the function \( 3^x \) by -1 reflects it over the x - axis. So the range of \( y = - 3^x \) will be the reflection of the range of \( 3^x \) over the x - axis. Since the range of \( 3^x \) is \( (0, \infty) \), the range of \( -3^x \) is \( (-\infty, 0) \) (because when we multiply each positive number \( y = 3^x>0 \) by - 1, we get \( y=-3^x < 0 \)).

Next, consider the transformation \( y=-3^x - 7 \). This is a vertical shift of the function \( y = - 3^x \) down by 7 units. To find the range of \( y=-3^x - 7 \), we take the range of \( y=-3^x \) (which is \( (-\infty, 0) \)) and subtract 7 from each value in that range. If we have \( y=-3^x < 0 \), then \( y=-3^x - 7<0 - 7=-7 \). So as \( x \to -\infty \), \( -3^x\to0 \), so \( -3^x - 7\to - 7 \), and as \( x \to \infty \), \( -3^x\to-\infty \), so \( -3^x - 7\to-\infty \). Therefore, the range of \( f(x)=-3^x - 7 \) is \( (-\infty, - 7) \).

Answer:

D. Domain \((-\infty, \infty)\), Range \((-\infty, -7)\)