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$Find each property of y=3*2^(x) **Type "infinity" for oo and "-infinity" for -oo. Write "DNE" if a property does not exist. Keep answers as fractions. Domain: (-oo,oo) Range: ( ◻ ◻ ) Horizontal Asymptote: y= ◻ X-Intercept: ◻ ,0) Y-intercept: (0, ◻ End Behavior: As x approaches infinity, y approaches ◻ @ As x approaches -infinity, y approaches ◻$

Find each property of $y=3 \cdot 2^{x}$ $\newline$ **Type

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Find equations of tangent lines using limits

Full solution

Q. Find each property of

y=3 \cdot 2^{x}

\newline

**Type

Identify Domain: Identify the domain of $y=3\cdot2^{x}$ . Since the function involves an exponential term $2^{x}$ , which is defined for all real numbers, the domain is all real numbers.
Identify Range: Identify the range of $y=3\cdot2^{x}$ . The exponential function $2^{x}$ takes positive values for all $x$ , and multiplying by $3$ scales these values but keeps them positive. Therefore, the range starts from the smallest value, $0$ (not included), to infinity.
Horizontal Asymptote: Determine the horizontal asymptote of $y=3\cdot2^{x}$ . As $x$ approaches $-\infty$ , $2^{x}$ approaches $0$ , and thus $3\cdot2^{x}$ approaches $0$ . Therefore, the horizontal asymptote is $y=0$ .
Find x-Intercept: Find the x-intercept of $y=3\cdot2^{x}$ . Set $y$ to $0$ and solve for $x$ : $\newline$ $0 = 3\cdot2^{x}$ $\newline$ Dividing both sides by $3$ gives $0 = 2^{x}$ . This equation has no solution because $2^{x} > 0$ for all $x$ . Thus, there is no x-intercept.
Calculate y-Intercept: Calculate the y-intercept of $y=3\cdot2^{x}$ . Set $x$ to $0$ and substitute into the equation: $\newline$ $y = 3\cdot2^{0} = 3\cdot1 = 3$ $\newline$ Thus, the y-intercept is $(0, 3)$ .
Analyzing End Behavior: Analyze the end behavior of $y=3\cdot2^{x}$ . As $x$ approaches infinity, $2^{x}$ approaches infinity, and thus $3\cdot2^{x}$ also approaches infinity. As $x$ approaches $-\infty$ , $2^{x}$ approaches $0$ , and thus $3\cdot2^{x}$ approaches $0$ .

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