Question

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

Answer 1

To analyze the functions \( f(x) = 9\left(\frac{3}{8}\right)^x \) and \( g(x) = 9\left(\frac{8}{3}\right)^x \), let's consider the properties of each function.

1. **Intercepts**:
- The **y-intercepts** of both functions can be found by evaluating them at \( x=0 \):
- \( f(0) = 9\left(\frac{3}{8}\right)^0 = 9 \)
- \( g(0) = 9\left(\frac{8}{3}\right)^0 = 9 \)
- Therefore, both functions have the same **y-intercept at (0, 9)**.

2. **X-Intercepts**:
- The **x-intercepts** occur when the function value is zero. However, for exponential functions, the output never actually reaches zero; they only get very close to zero but never cross the x-axis.
- So, both functions do not have an x-intercept.

3. **Increasing or Decreasing**:
- The function \( f(x) = 9\left(\frac{3}{8}\right)^x \) is a decreasing function because \( \frac{3}{8} < 1 \).
- The function \( g(x) = 9\left(\frac{8}{3}\right)^x \) is an increasing function because \( \frac{8}{3} > 1 \).

4. **Asymptotes**:
- Both functions approach an asymptote of \( y = 0 \) as \( x \) approaches infinity, since both functions are defined for all \( x \) and tend to 0 as \( x \) goes to positive or negative infinity.

From this analysis, the correct statements are:

- **c**: They have the same y-intercept.
- **f**: They both have an asymptote at \( y = 0 \).

Thus, the two correct answers are **c** and **f**.