Question

the function $f(x) = 2(3.5)^x$ is reflected across the $x$-axis to create $g(x)$. what is the function definition of $g(x)$? $g(x) = \square (3.5^x)$ what is $-2$ $1/2$ $2$ value of $g(x)$? what are the outputs for inputs of $-1$ and $1$ in $g(x)$? $g(-1) = \square$ $g(1) = \square$

Explanation:

Step1: Recall reflection over x - axis rule

Reflecting a function \(y = f(x)\) across the \(x\) - axis gives the new function \(y=-f(x)\).
Given \(f(x)=2(3.5)^{x}\), so \(g(x)=-f(x)=- 2(3.5)^{x}\).

Step2: Find \(g(-1)\)

Substitute \(x = - 1\) into \(g(x)=-2(3.5)^{x}\).
We know that \(a^{-n}=\frac{1}{a^{n}}\), so \((3.5)^{-1}=\frac{1}{3.5}=\frac{2}{7}\approx0.2857\).
Then \(g(-1)=-2\times(3.5)^{-1}=-2\times\frac{2}{7}=-\frac{4}{7}\approx - 0.5714\) (or we can calculate directly: \(g(-1)=-2\times(3.5)^{-1}=-2\div3.5=-\frac{2}{\frac{7}{2}}=-\frac{4}{7}\)).

Step3: Find \(g(1)\)

Substitute \(x = 1\) into \(g(x)=-2(3.5)^{x}\).
\(g(1)=-2\times(3.5)^{1}=-2\times3.5=-7\).

Answer:

(for the function definition of \(g(x)\)):
\(g(x)=-2(3.5^{x})\)

For \(g(-1)\):
\(g(-1)=-\frac{4}{7}\) (or approximately \(-0.571\))

For \(g(1)\):
\(g(1)=-7\)