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$f(x)=9^{x},g(x)=(\\frac {1}{9})^{x}$\n\nc. the y - intercept(s) of $f(x)$ are and $g(x)$ has no y - intercepts.\n(type an ordered pair, 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 $y=$ and the graph of $g(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nb. the graph of $f(x)$ has a horizontal asymptote at $y=$ and the graph of $g(x)$ has a horizontal asymptote at $y=$. (type integers or fractions.)\nc. the graph of $g(x)$ has a horizontal asymptote at $y=$ 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 intercepts, and asymptotes. identify the functions as increasing or decreasing.\n\n$f(x)=9^{x},g(x)=(\\frac {1}{9})^{x}$\n\nc. the y - intercept(s) of $f(x)$ are and $g(x)$ has no y - intercepts.\n(type an ordered pair, 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 $y=$ and the graph of $g(x)$ has no horizontal asymptote.\n(type an integer or a fraction.)\nb. the graph of $f(x)$ has a horizontal asymptote at $y=$ and the graph of $g(x)$ has a horizontal asymptote at $y=$. (type integers or fractions.)\nc. the graph of $g(x)$ has a horizontal asymptote at $y=$ 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: Find the y - intercept

For any function (y = a^{x}), the y - intercept is found by setting (x = 0). For (f(x)=9^{x}), when (x = 0), (f(0)=9^{0}=1). For (g(x)=\left(\frac{1}{9}\right)^{x}), when (x = 0), (g(0)=\left(\frac{1}{9}\right)^{0}=1). So both (f(x)) and (g(x)) have a y - intercept at ((0,1)).

Step2: Find the horizontal asymptote

For an exponential function of the form (y = a^{x}), where (a>0,a\neq1), the horizontal asymptote is (y = 0). For (f(x)=9^{x}), as (x\rightarrow-\infty), (y = 9^{x}\rightarrow0). For (g(x)=\left(\frac{1}{9}\right)^{x}=9^{-x}), as (x\rightarrow\infty), (y=\left(\frac{1}{9}\right)^{x}\rightarrow0).

Answer:

For the y - intercept part:

  • The y - intercept of (f(x)) is ((0,1)) and the y - intercept of (g(x)) is ((0,1)) (neither option C nor D is correct). For the horizontal asymptote part:
  • B. 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).