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\ncompare the two graphs. choose the correct answer below.\na. f and g have the same horizontal asymptote, but different y - intercepts. the functions both decay, but with g decaying faster than f. the functions have the same domain and range.\nb. f and g have the same x - intercept(s) and horizontal asymptote. g is a reflection of f across the x - axis, so the functions change at the same rate, though g decays and f grows. the functions have the same domain and range.\nc. f and g have the same y - intercept(s) and horizontal asymptote. since g is a reflection of f across the y - axis, g decays where f grows. the functions have the same domain and range.\nd. f and g have the same x - intercept(s) and horizontal asymptote. g is a reflection of f across the x - axis, so the functions change at the same rate, though g grows and f decays. the functions have the same domain and range.\ne. f and g have the same horizontal asymptote, but different y - intercepts. the functions both grow, but with g growing faster than f. the functions have the same domain and range.\nf. f and g have the same y - intercept(s) and horizontal asymptote. since g is a reflection of f across the y - axis, g grows where f decays. the functions have the same domain and range.

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\ncompare the two graphs. choose the correct answer below.\na. f and g have the same horizontal asymptote, but different y - intercepts. the functions both decay, but with g decaying faster than f. the functions have the same domain and range.\nb. f and g have the same x - intercept(s) and horizontal asymptote. g is a reflection of f across the x - axis, so the functions change at the same rate, though g decays and f grows. the functions have the same domain and range.\nc. f and g have the same y - intercept(s) and horizontal asymptote. since g is a reflection of f across the y - axis, g decays where f grows. the functions have the same domain and range.\nd. f and g have the same x - intercept(s) and horizontal asymptote. g is a reflection of f across the x - axis, so the functions change at the same rate, though g grows and f decays. the functions have the same domain and range.\ne. f and g have the same horizontal asymptote, but different y - intercepts. the functions both grow, but with g growing faster than f. the functions have the same domain and range.\nf. f and g have the same y - intercept(s) and horizontal asymptote. since g is a reflection of f across the y - axis, g grows where f decays. the functions have the same domain and range.

Answer

Brief Explanations:

  • For (y = a^{x}), the domain is ((-\infty,\infty)), range is ((0,\infty)), horizontal asymptote is (y = 0), and (y)-intercept is when (x = 0). For (f(x)=9^{x}), when (x = 0), (y=9^{0}=1). For (g(x)=\left(\frac{1}{9}\right)^{x}=9^{-x}), when (x = 0), (y=\left(\frac{1}{9}\right)^{0}=1). So they have the same (y)-intercept ((y = 1)) and horizontal asymptote ((y = 0)).
  • The function (y = a^{x}) is increasing if (a>1) (here (a = 9) for (f(x))) and decreasing if (0 < a<1) (here (a=\frac{1}{9}) for (g(x))). Also, (g(x)=\left(\frac{1}{9}\right)^{x}=9^{-x}), which is a reflection of (f(x)=9^{x}) across the (y)-axis ((y = a^{x}) and (y=a^{-x}) are reflections across the (y)-axis).

Answer:

C. (f) and (g) have the same (y)-intercept(s) and horizontal asymptote. Since (g) is a reflection of (f) across the (y)-axis, (g) decays where (f) grows. The functions have the same domain and range.