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.

the domain of f(x) is \boxed{} and the domain of g(x) is \boxed{}. the range of f(x) is \boxed{} and the range of g(x) is \boxed{}.(type your answer in interval notation.)

Explanation:

Step1: Domain of exponential functions

For any exponential function of the form \(y = a\cdot b^{x}+c\) (where \(a
eq0\), \(b> 0,b
eq1\)), the domain is all real numbers.
For \(f(x)=10^{x}\), the domain is \((-\infty,\infty)\) because we can substitute any real - number \(x\) into the function.
For \(g(x)=-10^{x}\), the domain is also \((-\infty,\infty)\) since we can substitute any real - number \(x\) into the function.

Step2: Range of \(y = 10^{x}\)

We know that for the exponential function \(y = 10^{x}\), since \(10^{x}>0\) for all \(x\in R\). So the range of \(f(x)=10^{x}\) is \((0,\infty)\)

Step3: Range of \(y=-10^{x}\)

Let \(t = 10^{x}\), then \(g(x)=-t\). Since \(t>0\), then \(-t<0\). So the range of \(g(x)=-10^{x}\) is \((-\infty,0)\)

Answer:

The domain of \(f(x)\) is \((-\infty,\infty)\) and the domain of \(g(x)\) is \((-\infty,\infty)\). The range of \(f(x)\) is \((0,\infty)\) and the range of \(g(x)\) is \((-\infty,0)\)