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\ncompare the two graphs. choose the correct answer below.\n\na. 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.\nb. 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.\nc. 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.\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 decays and f grows. the functions have the same domain and range.\ne. f and g have the same y - intercept(s) and horizontal asymptote. g is a reflection of f across the y - axis, so the functions change at the same rate, though g decays and f grows. the functions have the same domain and range.\nf. f and g have the same y - intercept(s) and horizontal asymptote. g is a reflection of f across the y - axis, so the functions change at the same rate, though g grows and 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\n$f(x)=(\\frac{7}{12})^{x},g(x)=(\\frac{12}{7})^{x}$\n\ncompare the two graphs. choose the correct answer below.\n\na. 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.\nb. 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.\nc. 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.\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 decays and f grows. the functions have the same domain and range.\ne. f and g have the same y - intercept(s) and horizontal asymptote. g is a reflection of f across the y - axis, so the functions change at the same rate, though g decays and f grows. the functions have the same domain and range.\nf. f and g have the same y - intercept(s) and horizontal asymptote. g is a reflection of f across the y - axis, so the functions change at the same rate, though g grows and f decays. the functions have the same domain and range.

Answer

Explanation:

Step1: Analyze the general form of exponential functions

The general form of an exponential function is (y = a\cdot b^{x}), where (a) is the (y -)intercept ((y=a) when (x = 0)) and (b) determines growth ((b>1)) or decay ((0 < b<1)). For (y = f(x)=\left(\frac{7}{12}\right)^{x}), (b=\frac{7}{12}\in(0,1)), so (f(x)) is a decay function. For (y = g(x)=\left(\frac{12}{7}\right)^{x}), (b = \frac{12}{7}>1), so (g(x)) is a growth function. When (x = 0), (f(0)=\left(\frac{7}{12}\right)^{0}=1) and (g(0)=\left(\frac{12}{7}\right)^{0}=1), so they have the same (y -)intercept ((y = 1)). The horizontal asymptote of an exponential function (y = b^{x}+k) (in our case (k = 0)) is (y = 0). The domain of (y=b^{x}) is ((-\infty,\infty)) and the range is ((0,\infty)) for both (y=\left(\frac{7}{12}\right)^{x}) and (y=\left(\frac{12}{7}\right)^{x}). Also, note that (g(x)=\left(\frac{12}{7}\right)^{x}=\left(\frac{7}{12}\right)^{-x}), which means (g(x)) is a reflection of (f(x)) across the (y -)axis ((y = f(-x)) is a reflection of (y = f(x)) across the (y -)axis).

Step2: Check each option

  • Option A: Incorrect, because (f) decays and (g) grows.
  • Option B: Incorrect, because (f) decays and (g) grows.
  • Option C: Incorrect, because they have the same (y -)intercept (not (x -)intercept, since (y=b^{x}) has no (x -)intercept for (b>0,b\neq1))
  • Option D: Incorrect, because they have the same (y -)intercept (not (x -)intercept, since (y=b^{x}) has no (x -)intercept for (b>0,b\neq1))
  • Option E: Incorrect, because (g) grows and (f) decays.
  • Option F: Correct. They have (y(0) = 1) (same (y -)intercept), horizontal asymptote (y = 0), (g(x)=f(-x)) (reflection across the (y -)axis), domain ((-\infty,\infty)) and range ((0,\infty)), (f) decays ((\frac{7}{12}<1)) and (g) grows ((\frac{12}{7}>1))

Answer:

F. (f) and (g) have the same (y -)intercept(s) and horizontal asymptote. (g) is a reflection of (f) across the (y -)axis, so the functions change at the same rate, though (g) grows and (f) decays. The functions have the same domain and range.