the equations of two exponential functions are shown.

Question

f(x) = 9(3/8)^x
g(x) = 9(8/3)^x
which two statements are true about the graphs of the functions
a. they have the same slope.
b. they have the same x-intercept.
c. they have thesame y-intercept.
d. they are both decreasing functions.
e. they are both increasing functions.
f. they both have an asymptote at y = 0
select TWO correct answers

Answer 1

To analyze the equations of the two exponential functions \( f(x) = 9\left(\frac{3}{8}\right)^x \) and \( g(x) = 9\left(\frac{8}{3}\right)^x \), we can make the following observations:

1. Y-Intercept:

   • At \( x = 0 \), for both functions: \[ f(0) = 9\left(\frac{3}{8}\right)^0 = 9 \] \[ g(0) = 9\left(\frac{8}{3}\right)^0 = 9 \]
   • Both functions have the same y-intercept which is \( 9 \).

2. Slope:

   • The concept of "slope" in the context of exponential functions isn't well-defined like in linear functions. In these functions:
     • \( f(x) \) is a decreasing function as \( \frac{3}{8} < 1 \).
     • \( g(x) \) is an increasing function as \( \frac{8}{3} > 1 \).
   • Therefore, they do not have the same slope.

3. X-Intercept:

   • \( f(x) \) has no x-intercept since \( \left(\frac{3}{8}\right)^x > 0 \) for all \( x \).
   • \( g(x) \) has an x-intercept (it goes to infinity as \( x \) approaches negative values).
   • So they do not have the same x-intercept.

4. Behavior of Functions:

   • \( f(x) \) is decreasing while \( g(x) \) is increasing (as explained earlier).
   • Thus, neither function is increasing or both decreasing.

5. Asymptote:

   • Both functions approach \( y = 0 \) as \( x \) approaches infinity. They both have an asymptote at \( y = 0 \).

Based on this analysis, the two correct statements are:

• c. They have the same y-intercept.
• f. They both have an asymptote at y = 0.