(1 point)

Describe how $g(x)$ represents a transformation of the function $f(x)$ :
- $g(x)=9 f(-x)$
vertical stretch and reflection across $y$-axis
vertical stretch and reflection across $x$-axis
vertical compression and reflection across $y$-axis
vertical compression and reflection across $\mathrm{x}$-axis
- $g(x)=-f\left(\frac{1}{5} x\right)$
horizontal stretch and reflection across $y$-axis
horizontal stretch and reflection across $\mathrm{x}$-axis
horizontal compression and reflection across $y$-axis
horizontal compression and reflection across $\mathrm{x}$-axis

Step 1 ：The function \(g(x)=9f(-x)\) represents a transformation of the function \(f(x)\) in two ways:

Step 2 ：The factor of 9 in front of \(f(-x)\) indicates a vertical stretch by a factor of 9. This means that the graph of \(f(x)\) is stretched vertically by a factor of 9 to obtain the graph of \(g(x)\).

Step 3 ：The negative sign inside the function \(f(-x)\) indicates a reflection across the y-axis. This means that the graph of \(f(x)\) is reflected across the y-axis to obtain the graph of \(g(x)\).

Step 4 ：So, the correct answer is \(\boxed{\text{'vertical stretch and reflection across y-axis'}}\).

Step 5 ：The function \(g(x)=-f\left(\frac{1}{5} x\right)\) also represents a transformation of the function \(f(x)\) in two ways:

Step 6 ：The negative sign in front of \(f\left(\frac{1}{5} x\right)\) indicates a reflection across the x-axis. This means that the graph of \(f(x)\) is reflected across the x-axis to obtain the graph of \(g(x)\).

Step 7 ：The factor of \(\frac{1}{5}\) inside the function \(f\left(\frac{1}{5} x\right)\) indicates a horizontal stretch by a factor of 5. This means that the graph of \(f(x)\) is stretched horizontally by a factor of 5 to obtain the graph of \(g(x)\).

Step 8 ：So, the correct answer is \(\boxed{\text{'horizontal stretch and reflection across x-axis'}}\).