Question

using your graphing calculator, investigate the graphs of the following family (group) of functions. describe any relationships within the family, including domain and range, growth or decay factors, vertical and horizontal intercepts, and asymptotes. identify the functions as increasing or decreasing.

$f(x)=10^{x},g(x)=-10^{x}$

graph $f(x)$ and $g(x)$ in the same viewing window.

all graphs are displayed in a $-5,5,1$ by $-12,12,1$ viewing window.

Explanation:

Step1: Analyze the function \( f(x) = 10^{x} \)

For \( y = 10^{x} \), the domain is \( (-\infty,\infty) \), the range is \( (0,\infty) \). When \( x = 0 \), \( y=10^{0}=1 \), so the \( y - \)intercept is \( (0,1) \). There is no \( x - \)intercept. As \( x
ightarrow-\infty \), \( y = 10^{x}
ightarrow0 \) (horizontal asymptote \( y = 0 \)), and as \( x
ightarrow\infty \), \( y = 10^{x}
ightarrow\infty \). The function \( y = 10^{x} \) is an increasing function.

Step2: Analyze the function \( g(x)=-10^{x} \)

For \( y=-10^{x} \), the domain is \( (-\infty,\infty) \), the range is \( (-\infty,0) \). When \( x = 0 \), \( y=-10^{0}=-1 \), so the \( y - \)intercept is \( (0, - 1) \). There is no \( x - \)intercept. As \( x
ightarrow-\infty \), \( y=-10^{x}
ightarrow0 \) (horizontal asymptote \( y = 0 \)), and as \( x
ightarrow\infty \), \( y=-10^{x}
ightarrow-\infty \). The function \( y=-10^{x} \) is a decreasing function. Also, \( g(x)=-f(x) \), which means the graph of \( g(x) \) is the reflection of the graph of \( f(x) \) about the \( x - \)axis.

If we consider the general shape of exponential functions \( y = a\cdot b^{x}\) (\(b>1\) for growth). For \( f(x)=10^{x}\), \(a = 1,b = 10\) (growth, above \(x - \)axis) and for \(g(x)=-10^{x}\), \(a=-1,b = 10\) (reflection of \(f(x)\) over \(x - \)axis, below \(x - \)axis)

Answer:

B.