using your graphing calculator, investigate the graphs of the following family (group) of functions…

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.\n\n$f(x)=(\\frac{7}{12})^{x},g(x)=(\\frac{12}{7})^{x}$\n\ngraph $f(x)$ and $g(x)$ in the same viewing window.\n\nall graphs are displayed in a $-5,5,1$ by $-12,12,1$ viewing window.\n\nthe domain of $f(x)$ is $\\square$ and the domain of $g(x)$ is $\\square$. the range of $f(x)$ is $\\square$ and the range of $g(x)$ is $\\square$.\n\n(type your answer in interval notation.)

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.\n\n$f(x)=(\\frac{7}{12})^{x},g(x)=(\\frac{12}{7})^{x}$\n\ngraph $f(x)$ and $g(x)$ in the same viewing window.\n\nall graphs are displayed in a $-5,5,1$ by $-12,12,1$ viewing window.\n\nthe domain of $f(x)$ is $\\square$ and the domain of $g(x)$ is $\\square$. the range of $f(x)$ is $\\square$ and the range of $g(x)$ is $\\square$.\n\n(type your answer in interval notation.)

Answer

Explanation:

Step1: Analyze the domain of exponential functions

For any exponential function (y = a^{x}) ((a>0,a\neq1)), the domain is all real numbers. So for (f(x)=\left(\frac{7}{12}\right)^{x}) and (g(x)=\left(\frac{12}{7}\right)^{x}), the domain of (f(x)) is ((-\infty,\infty)) and the domain of (g(x)) is ((-\infty,\infty)).

Step2: Analyze the range of exponential functions

For the exponential function (y = a^{x}) ((a>0,a\neq1)):

  • When (a> 1) (for (g(x)=\left(\frac{12}{7}\right)^{x}), (a = \frac{12}{7}>1)), as (x\rightarrow-\infty), (y\rightarrow0) and as (x\rightarrow\infty), (y\rightarrow\infty).
  • When (0 < a<1) (for (f(x)=\left(\frac{7}{12}\right)^{x}), (a=\frac{7}{12}\in(0,1))), as (x\rightarrow-\infty), (y\rightarrow\infty) and as (x\rightarrow\infty), (y\rightarrow0). The range of an exponential function (y = a^{x}) ((a>0,a\neq1)) is ((0,\infty)). So the range of (f(x)) is ((0,\infty)) and the range of (g(x)) is ((0,\infty)).

Answer:

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