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$f(x)=(\\frac{7}{12})^x,g(x)=(\\frac{12}{7})^x$\n(type your answer in interval notation.)\nthe graph of $f(x)$ has a decay factor of $\\frac{7}{12}$. the graph of $g(x)$ has a growth factor of $\\frac{12}{7}$.\n(type integers or fractions.)\nidentify the $x$-intercept(s) of $f(x)$ and $g(x)$. select the correct choice below and, if necessary, fill in any answer box(es) to complete your choice.\na. the $x$-intercept(s) of $f(x)$ are $\\square$ and $g(x)$ has no $x$-intercepts.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nb. the $x$-intercept(s) of $g(x)$ are $\\square$ and $f(x)$ has no $x$-intercepts.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nc. the $x$-intercept(s) of $f(x)$ are $\\square$ and the $x$-intercept(s) of $g(x)$ are $\\square$.\n(type ordered pairs, using integers or fractions. use a comma to separate answers as needed.)\nd. $f(x)$ and $g(x)$ have no $x$-intercepts.

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$f(x)=(\\frac{7}{12})^x,g(x)=(\\frac{12}{7})^x$\n(type your answer in interval notation.)\nthe graph of $f(x)$ has a decay factor of $\\frac{7}{12}$. the graph of $g(x)$ has a growth factor of $\\frac{12}{7}$.\n(type integers or fractions.)\nidentify the $x$-intercept(s) of $f(x)$ and $g(x)$. select the correct choice below and, if necessary, fill in any answer box(es) to complete your choice.\na. the $x$-intercept(s) of $f(x)$ are $\\square$ and $g(x)$ has no $x$-intercepts.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nb. the $x$-intercept(s) of $g(x)$ are $\\square$ and $f(x)$ has no $x$-intercepts.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nc. the $x$-intercept(s) of $f(x)$ are $\\square$ and the $x$-intercept(s) of $g(x)$ are $\\square$.\n(type ordered pairs, using integers or fractions. use a comma to separate answers as needed.)\nd. $f(x)$ and $g(x)$ have no $x$-intercepts.

Answer

Explanation:

Step1: Recall the property of exponential functions

For an exponential function (y = a^x) ((a>0,a\neq1)), when (0 < a<1), (y=a^x) is a decay function; when (a > 1), (y=a^x) is a growth function. And the equation (a^x=0) has no real - solution for any real (x) since (a^x>0) for all (x\in R) when (a>0,a\neq1).

Step2: Analyze (f(x)=(\frac{7}{12})^x) and (g(x)=(\frac{12}{7})^x)

For (f(x)=(\frac{7}{12})^x), since (0<\frac{7}{12}<1), it is a decay function. For (g(x)=(\frac{12}{7})^x), since (\frac{12}{7}>1), it is a growth function. To find the (x) - intercept, we set (y = 0). For (f(x)): Solve the equation ((\frac{7}{12})^x=0). Let (y = (\frac{7}{12})^x). Taking the natural logarithm of both sides, (\ln y=x\ln(\frac{7}{12})). Since ((\frac{7}{12})^x>0) for all (x\in R) (because the exponential function (y = a^x,a>0,a\neq1) has a range of ((0,\infty))), there is no real (x) such that ((\frac{7}{12})^x = 0). For (g(x)): Solve the equation ((\frac{12}{7})^x=0). Let (y = (\frac{12}{7})^x). Taking the natural logarithm of both sides, (\ln y=x\ln(\frac{12}{7})). Since ((\frac{12}{7})^x>0) for all (x\in R) (because the exponential function (y = a^x,a>0,a\neq1) has a range of ((0,\infty))), there is no real (x) such that ((\frac{12}{7})^x=0).

Answer:

D. (f(x)) and (g(x)) have no (x) - intercepts.