Question
Luis draws this graph of the exponential function f(x) =
1
2
(5)
x − 3. Which inequality shows the
range of Luis’s function?
(1 point)
y < − 3
y > − 3
−∞ < y < ∞
−3 < y < 10
1
2
(5)
x − 3. Which inequality shows the
range of Luis’s function?
(1 point)
y < − 3
y > − 3
−∞ < y < ∞
−3 < y < 10
Answers
GPT-4o mini
To determine the range of the function \( f(x) = \frac{1}{2}(5^x) - 3 \), we can analyze the behavior of the exponential component.
1. The term \( 5^x \) is an exponential function that approaches \( 0 \) as \( x \to -\infty \) and increases without bound as \( x \to \infty \).
2. Since this function is multiplied by \( \frac{1}{2} \), the output of \( \frac{1}{2}(5^x) \) will also approach \( 0 \) as \( x \to -\infty \) and increase without bound as \( x \to \infty \).
3. Finally, subtracting \( 3 \) from the function shifts the entire graph down by \( 3 \) units. Therefore, as \( x \to -\infty \), \( f(x) \) approaches \( -3 \).
4. As \( x \) increases, \( f(x) \) will eventually become larger than \( -3 \) and continue to rise indefinitely.
Thus, the range of the function \( f(x) \) is:
\[
y > -3
\]
So the correct answer is:
**y > -3**
1. The term \( 5^x \) is an exponential function that approaches \( 0 \) as \( x \to -\infty \) and increases without bound as \( x \to \infty \).
2. Since this function is multiplied by \( \frac{1}{2} \), the output of \( \frac{1}{2}(5^x) \) will also approach \( 0 \) as \( x \to -\infty \) and increase without bound as \( x \to \infty \).
3. Finally, subtracting \( 3 \) from the function shifts the entire graph down by \( 3 \) units. Therefore, as \( x \to -\infty \), \( f(x) \) approaches \( -3 \).
4. As \( x \) increases, \( f(x) \) will eventually become larger than \( -3 \) and continue to rise indefinitely.
Thus, the range of the function \( f(x) \) is:
\[
y > -3
\]
So the correct answer is:
**y > -3**