Pregunta
State the horizontal asymptote for the function \( f(x)=2^{12 x}-5 \) : \( y=2 \) \( x=2 \) \( y=-5 \) \( x=-5 \)
Ask by Macdonald Sandoval. in the United States
Jan 25,2025
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The horizontal asymptote is \( y = -5 \).
Solución
To determine the horizontal asymptote of the function \( f(x) = 2^{12x} - 5 \), let's analyze its behavior as \( x \) approaches both positive and negative infinity.
1. **As \( x \to \infty \):**
- \( 2^{12x} \) grows exponentially towards infinity.
- Therefore, \( f(x) = 2^{12x} - 5 \) also tends to infinity.
2. **As \( x \to -\infty \):**
- \( 2^{12x} \) approaches \( 0 \) because the exponent is negative and large.
- Thus, \( f(x) = 2^{12x} - 5 \) approaches \( -5 \).
Since the function approaches \( -5 \) as \( x \) approaches negative infinity, the **horizontal asymptote** of the function is \( y = -5 \).
**Correct Answer:** \( y = -5 \)
Revisado y aprobado por el equipo de tutoría de UpStudy
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For the function \( f(x) = 2^{12x} - 5 \), as \( x \) approaches negative infinity, the term \( 2^{12x} \) approaches 0. Thus, \( f(x) \) approaches \( -5 \). Therefore, the horizontal asymptote is \( y = -5 \). Another interesting aspect of horizontal asymptotes is that they indicate the long-term behavior of a function as \( x \) moves towards infinity or negative infinity. In the case of exponential functions, such as this one, they can reveal how the function behaves with very large or very small values of \( x \), providing insights into its growth or decay patterns.
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