Question
An exponential function is described by the equation h(x)=49(2/7)^x
. Which inequality describes the range of this function?
(1 point)
Responses
0<y<16
0 less than y less than 16
y≥0
y greater-than-or-equal-to 0
−∞<y<∞
negative infinity less than y less than infinity
y>0
. Which inequality describes the range of this function?
(1 point)
Responses
0<y<16
0 less than y less than 16
y≥0
y greater-than-or-equal-to 0
−∞<y<∞
negative infinity less than y less than infinity
y>0
Answers
Answered by
GPT-4o mini
To determine the range of the function \( h(x) = 49 \left( \frac{2}{7} \right)^x \), we need to analyze the behavior of this exponential function.
1. The base of the exponent, \( \frac{2}{7} \), is between 0 and 1. This means the function \( \left( \frac{2}{7} \right)^x \) will decrease as \( x \) increases.
2. As \( x \) approaches positive infinity, \( h(x) \) approaches 0, but it will never actually reach 0 (the function goes to 0 asymptotically).
3. As \( x \) approaches negative infinity, \( h(x) \) increases without bound, meaning it can take any positive value.
Based on this analysis, the range of \( h(x) \) is \( 0 < y < \infty \).
However, since the inequality does not include 0, the appropriate choice from the provided options is:
**y > 0**
1. The base of the exponent, \( \frac{2}{7} \), is between 0 and 1. This means the function \( \left( \frac{2}{7} \right)^x \) will decrease as \( x \) increases.
2. As \( x \) approaches positive infinity, \( h(x) \) approaches 0, but it will never actually reach 0 (the function goes to 0 asymptotically).
3. As \( x \) approaches negative infinity, \( h(x) \) increases without bound, meaning it can take any positive value.
Based on this analysis, the range of \( h(x) \) is \( 0 < y < \infty \).
However, since the inequality does not include 0, the appropriate choice from the provided options is:
**y > 0**