Let the function f be an exponential function represented by f[x) = a(by - c, where a = 0, 0 ≤ b < 1, and c is a real number. Describe the end behavior of the function using the mathematical notation of a lim
• The lim
f(x) = - C and the
x+ - 30
lim f(x) = co
O The
lim
f(x) = ∞ and the lim f(x) = c
x→ o
O The
lim
{x) = C and the lim F{x) = 00
x → o
• The
lim f[x) = 00
and the
lim f(x) = - c

To analyze the **end behavior** of the function \( f(x) = a \cdot b^x - c \), where \( a > 0 \), \( 0 \leq b < 1 \), and \( c \) is a real number, we compute the limits of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \):

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### 1. Behavior as \( x \to \infty \):
Since \( b \) is between 0 and 1, \( b^x \to 0 \) as \( x \to \infty \). Therefore:
\[
f(x) = a \cdot b^x - c \implies f(x) \to -c \text{ as } x \to \infty.
\]

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### 2. Behavior as \( x \to -\infty \):
When \( x \to -\infty \), \( b^x \to \infty \), because \( b < 1 \) and the reciprocal of a small number grows unbounded. Therefore:
\[
f(x) = a \cdot b^x - c \implies f(x) \to \infty \text{ as } x \to -\infty.
\]

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### Final Answer:
The correct description of the end behavior is:
\[
\lim_{x \to \infty} f(x) = -c \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty.
\]

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Do you have further questions, or should we explore more examples? Here are five related questions for practice:

1. What happens if \( a < 0 \) in the function \( f(x) = a \cdot b^x - c \)?
2. How does the graph of \( f(x) = a \cdot b^x - c \) change when \( c \) is increased or decreased?
3. What is the behavior of \( f(x) = a \cdot b^x - c \) when \( b = 1 \)?
4. Can the function \( f(x) = a \cdot b^x - c \) ever have horizontal asymptotes? If so, where?
5. How would \( f(x) = a \cdot b^x - c \) change if \( b > 1 \) instead of \( 0 < b < 1 \)?

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**Tip:** The exponential base \( b \) dictates the growth or decay behavior. For \( b < 1 \), the function represents exponential decay. For \( b > 1 \), it represents exponential growth.

Let the function f be an exponential function represented by f(x) = a(b^x) - c, where a > 0, 0 ≤ b < 1, and c is a real number. Describe the end behavior of the function using the mathematical notation of limits.

Understand the end behavior of the exponential function f(x) = a(b^x) - c, where a > 0 and 0 ≤ b < 1. Explore key mathematical concepts such as exponential decay, limit properties, and asymptotes. Suitable for students in Grades 11-12.

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