Math Snap

STEP 1

1. We are given three separate functions, each defined in terms of the exponential function $e^x$.
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 $f(x) = 3(e^x + e^{-x})$.
2. Analyze the function $f(w) = \frac{e^w + 2}{e^w}$.
3. Analyze the function $f(x) = 2e^{3x-1}$.

STEP 3

Consider the function $f(x) = 3(e^x + e^{-x})$.
- Recognize that this is a combination of exponential growth and decay terms.
- The term $e^x$ represents exponential growth, while $e^{-x}$ represents exponential decay.
- The function can be rewritten using hyperbolic cosine: $f(x) = 3 \cdot 2 \cosh(x)$, where $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
- This function is symmetric about the y-axis, as $f(-x) = f(x)$.

STEP 4

Consider the function $f(w) = \frac{e^w + 2}{e^w}$.
- Simplify the expression by dividing each term in the numerator by $e^w$.
- This results in: $f(w) = 1 + \frac{2}{e^w}$.
- As $w \to \infty$, $e^w \to \infty$ and $\frac{2}{e^w} \to 0$, so $f(w) \to 1$.
- As $w \to -\infty$, $e^w \to 0$ and $\frac{2}{e^w} \to \infty$, so $f(w) \to \infty$.

Consider the function $f(x) = 2e^{3x-1}$.
- Recognize that this is an exponential function with a transformation.
- The exponent $3x - 1$ indicates a horizontal stretch/compression and shift.
- Rewrite the exponent as $e^{3x} \cdot e^{-1}$, which simplifies to $\frac{2}{e} \cdot e^{3x}$.
- This function represents exponential growth, with a growth rate determined by the factor $3$.
Each function has been analyzed in terms of its structure and behavior.