Question

1 select the correct answer. consider the graph of the function $f(x)=e^{x}$. which statement describes a key feature of function $g$ if $g(x)=e^{x}-7$? a. range of $y < - 7$ b. horizontal asymptote of $y=-7$ c. y - intercept at $(0, - 7)$ d. domain of $x > - 7$

Explanation:

Step1: Recall properties of exponential - functions

The parent function is \(y = e^{x}\), which has a range of \(y>0\), a horizontal asymptote of \(y = 0\), a \(y\) - intercept at \((0,1)\), and a domain of all real numbers (\(x\in(-\infty,\infty)\)).

Step2: Analyze the function \(g(x)=e^{x}-7\)

For the function \(g(x)=e^{x}-7\), we use the transformation rules. The graph of \(y = f(x)-k\) is a vertical shift of the graph of \(y = f(x)\) down by \(k\) units.

• Domain: The domain of an exponential function \(y = a^{x}\) (where \(a>0,a

eq1\)) is all real numbers. So, the domain of \(g(x)=e^{x}-7\) is \(x\in(-\infty,\infty)\) since there are no restrictions on the input \(x\) for the exponential function \(e^{x}\).

• Range: Since the range of \(y = e^{x}\) is \(y>0\), for \(g(x)=e^{x}-7\), we subtract 7 from each value in the range of \(e^{x}\). So the range is \(y>-7\).
• Horizontal asymptote: The horizontal asymptote of \(y = e^{x}\) is \(y = 0\). When we shift the graph of \(y = e^{x}\) down 7 units to get \(y = e^{x}-7\), the horizontal asymptote also shifts down 7 units. So the horizontal asymptote of \(g(x)=e^{x}-7\) is \(y=-7\).
• \(y\) - intercept: To find the \(y\) - intercept, we set \(x = 0\). Then \(g(0)=e^{0}-7=1 - 7=-6\), so the \(y\) - intercept is \((0,-6)\).

Answer:

B. horizontal asymptote of \(y=-7\)