Question
What is the range of this exponential function?
y=−3(1.5)x
(1 point)
$$
Remaining Attempts : 3
y=−3(1.5)x
(1 point)
$$
Remaining Attempts : 3
Answers
Answered by
GPT-4o mini
To determine the range of the exponential function \( y = -3(1.5)^x \), we analyze the behavior of the function:
1. **Exponential Growth**: The term \( (1.5)^x \) represents exponential growth. As \( x \) increases, \( (1.5)^x \) increases, starting from \( 1 \) when \( x = 0 \) and approaching infinity as \( x \) approaches infinity.
2. **Transformation**: The function is multiplied by \(-3\). This means that the entire graph is reflected over the x-axis and multiplied by \( 3 \).
3. **Behavior of \( y \)**:
- When \( x \to -\infty \), \( (1.5)^x \to 0 \), so \( y \to -3 \times 0 = 0 \).
- When \( x \to \infty \), \( (1.5)^x \to \infty \), so \( y \to -3 \times \infty = -\infty \).
4. **Range**: Since the function approaches \( 0 \) but never actually reaches \( 0 \) (as it is always negative), and goes to \( -\infty \), we can conclude:
The range of the function \( y = -3(1.5)^x \) is \( (-\infty, 0) \).
1. **Exponential Growth**: The term \( (1.5)^x \) represents exponential growth. As \( x \) increases, \( (1.5)^x \) increases, starting from \( 1 \) when \( x = 0 \) and approaching infinity as \( x \) approaches infinity.
2. **Transformation**: The function is multiplied by \(-3\). This means that the entire graph is reflected over the x-axis and multiplied by \( 3 \).
3. **Behavior of \( y \)**:
- When \( x \to -\infty \), \( (1.5)^x \to 0 \), so \( y \to -3 \times 0 = 0 \).
- When \( x \to \infty \), \( (1.5)^x \to \infty \), so \( y \to -3 \times \infty = -\infty \).
4. **Range**: Since the function approaches \( 0 \) but never actually reaches \( 0 \) (as it is always negative), and goes to \( -\infty \), we can conclude:
The range of the function \( y = -3(1.5)^x \) is \( (-\infty, 0) \).
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