To analyze the two functions \( F(x) = 9(3/8)^x \) and \( G(x) = 9(8/3)^x \), we can look at their general properties:
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Y-Intercept: Both functions have the same y-intercept because when \( x = 0 \):
- \( F(0) = 9(3/8)^0 = 9 \)
- \( G(0) = 9(8/3)^0 = 9 \) Thus, both functions have the same y-intercept at \( (0, 9) \).
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X-Intercept: To find \( x \)-intercepts, we set \( F(x) \) and \( G(x) \) equal to zero. However, both exponential functions do not cross the x-axis, as they never equal zero for any real \( x \). Therefore, they do not have an x-intercept.
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Increasing or Decreasing Functions: The base of the exponential function for \( F(x) \) is \( \frac{3}{8} \), which is less than 1, indicating that \( F(x) \) is a decreasing function. Conversely, the base of \( G(x) \) is \( \frac{8}{3} \), which is greater than 1, indicating that \( G(x) \) is an increasing function.
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Slope: Exponential functions do not have a constant slope; their rates of growth change. Therefore, they do not have the same slope.
Based on this analysis, the two true statements about the graphs of the functions are:
- They have the same y intercept.
- They are both decreasing functions.
Hence, the correct answers are:
- They have the same y intercept.
- They are both decreasing functions.
1 answer