which function represents a vertical stretch of an exponential function?\n f(x)=3left(\frac{1}{2}\right)^x…

Explanation:

Step1: Recall vertical - stretch rule

For an exponential function $y = a\cdot b^{x}$, if $|a|> 1$, it represents a vertical stretch.

Step2: Analyze each option

• Option 1: $f(x)=3\left(\frac{1}{2}\right)^{x}$, here $a = 3>1$, it is an exponential function $y = a\cdot b^{x}$ with $b=\frac{1}{2}$ and $a = 3$, so it represents a vertical stretch.
• Option 2: $f(x)=\frac{1}{2}(3)^{x}$, here $a=\frac{1}{2}<1$, it represents a vertical compression.
• Option 3: $f(x)=(3)^{2x}=9^{x}$, it is in the form $y = b^{x}$ without a coefficient $a$ (equivalent to $a = 1$), no vertical - stretch or compression.
• Option 4: $f(x)=3^{\frac{1}{2}x}=\sqrt{3}^{x}$, it is in the form $y = b^{x}$ without a coefficient $a$ (equivalent to $a = 1$), no vertical - stretch or compression.

Answer:

$f(x)=3\left(\frac{1}{2}\right)^{x}$