Question

which function represents a reflection of $f(x) = 5(0.8)^x$ across the $x$-axis?
$g(x) = 5(-0.8)^x$
$g(x) = -5(0.8)^x$
$g(x) = \frac{1}{5}(0.8)^x$
$g(x) = 5(0.8)^{-x}$

Explanation:

Step1: Recall reflection over x - axis rule

The rule for reflecting a function \(y = f(x)\) across the \(x\) - axis is \(y=-f(x)\).

Step2: Apply the rule to \(f(x) = 5(0.8)^{x}\)

Given \(f(x)=5(0.8)^{x}\), when we reflect it across the \(x\) - axis, we replace \(f(x)\) with \(-f(x)\). So \(g(x)=-f(x)=- 5(0.8)^{x}\).
Let's analyze the other options:

• For \(g(x) = 5(-0.8)^{x}\), this is a reflection over the \(y\) - axis combined with other transformations (since \(y = f(-x)\) is reflection over \(y\) - axis, but here the base is \(- 0.8\), not just a sign change of the function value), so this is not a reflection over \(x\) - axis.
• For \(g(x)=\frac{1}{5}(0.8)^{x}\), this is a vertical compression of the function \(f(x)\), not a reflection.
• For \(g(x)=5(0.8)^{-x}\), this is a reflection over the \(y\) - axis (since \(y = f(-x)\) is reflection over \(y\) - axis), not over the \(x\) - axis.

Answer:

\(g(x)=-5(0.8)^{x}\) (the second option)