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 intercep and asymptotes. identify the functions as increasing or decreasing.\n\n$f(x)=(\\frac{7}{12})^x,g(x)=(\\frac{12}{7})^x$\n\nc. the y - intercept(s) of $f(x)$ are $(0,1)$ and the y - intercept(s) of $g(x)$ are $(0,1)$.\n(type ordered pairs, using integers or fractions. use a comma to separate answers as needed.)\nd. $f(x)$ and $g(x)$ have no y - intercepts.\n\nidentify any horizontal asymptotes. select the correct choice below and, if necessary, fill in any answer box(es) to complete your choice.\na. the graph of $f(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $g(x)$ has a horizontal asymptote at $y = \\square$. (type integers or fractions.)\nb. the graph of $f(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $g(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nc. the graph of $g(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $f(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nd. $f(x)$ and $g(x)$ have no horizontal asymptotes.

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 intercep and asymptotes. identify the functions as increasing or decreasing.\n\n$f(x)=(\\frac{7}{12})^x,g(x)=(\\frac{12}{7})^x$\n\nc. the y - intercept(s) of $f(x)$ are $(0,1)$ and the y - intercept(s) of $g(x)$ are $(0,1)$.\n(type ordered pairs, using integers or fractions. use a comma to separate answers as needed.)\nd. $f(x)$ and $g(x)$ have no y - intercepts.\n\nidentify any horizontal asymptotes. select the correct choice below and, if necessary, fill in any answer box(es) to complete your choice.\na. the graph of $f(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $g(x)$ has a horizontal asymptote at $y = \\square$. (type integers or fractions.)\nb. the graph of $f(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $g(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nc. the graph of $g(x)$ has a horizontal asymptote at $y = \\square$ and the graph of $f(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nd. $f(x)$ and $g(x)$ have no horizontal asymptotes.

Answer

Explanation:

Step1: Recall the form of exponential function

The general form of an exponential function is (y = a\cdot b^{x}), where (a\neq0), (b>0) and (b\neq1). For (y = b^{x}), when (b>1), the function is an increasing exponential function; when (0 < b<1), the function is a decreasing exponential function. The domain of (y = b^{x}) is ((-\infty,\infty)).

For (f(x)=\left(\frac{7}{12}\right)^{x}), since (0<\frac{7}{12}<1), (f(x)) is a decreasing function. For (g(x)=\left(\frac{12}{7}\right)^{x}), since (\frac{12}{7}>1), (g(x)) is an increasing function.

Step2: Find the horizontal asymptote

The horizontal asymptote of the exponential function (y = b^{x}) ((b>0,b\neq1)) is found by considering the limit as (x\rightarrow\pm\infty).

  • For (y = f(x)=\left(\frac{7}{12}\right)^{x}): (\lim_{x\rightarrow\infty}\left(\frac{7}{12}\right)^{x}=0) and (\lim_{x\rightarrow-\infty}\left(\frac{7}{12}\right)^{x}=\infty)
  • For (y = g(x)=\left(\frac{12}{7}\right)^{x}): (\lim_{x\rightarrow-\infty}\left(\frac{12}{7}\right)^{x}=0) and (\lim_{x\rightarrow\infty}\left(\frac{12}{7}\right)^{x}=\infty)

Answer:

A. The graph of (f(x)) has a horizontal asymptote at (y = 0) and the graph of (g(x)) has a horizontal asymptote at (y = 0)