Question\begin{align*}
1. & \quad f(x) = 3\left(e^{x} + e^{-x}\right) \\
3. & \quad f(w) = \frac{e^{w} + 2}{e^{w}} \\
5. & \quad f(x) = 2 e^{3x-1}
\end{align*}
Studdy Solution
STEP 1
1. We are given three separate functions, each defined in terms of the exponential function .
2. We need to analyze each function individually.
3. The goal is to understand the structure and behavior of each function.
STEP 2
1. Analyze the function .
2. Analyze the function .
3. Analyze the function .
STEP 3
Consider the function .
- Recognize that this is a combination of exponential growth and decay terms.
- The term represents exponential growth, while represents exponential decay.
- The function can be rewritten using hyperbolic cosine: , where .
- This function is symmetric about the y-axis, as .
STEP 4
Consider the function .
- Simplify the expression by dividing each term in the numerator by .
- This results in: .
- As , and , so .
- As , and , so .
STEP 5
Consider the function .
- Recognize that this is an exponential function with a transformation.
- The exponent indicates a horizontal stretch/compression and shift.
- Rewrite the exponent as , which simplifies to .
- This function represents exponential growth, with a growth rate determined by the factor .
Each function has been analyzed in terms of its structure and behavior.
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