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 $g(x)$? $g(x) = -2 (3.5^x)$ what is the initial value of $g(x)$? -3.5 the outputs for inputs of -1 and 1 in $g(x)$? -0.57 $g(1) = -7$

Explanation:

Step1: Reflect across x - axis

To reflect a function \(y = f(x)\) across the \(x\) - axis, we use the transformation \(g(x)=-f(x)\). Given \(f(x) = 2(3.5)^{x}\), then \(g(x)=-2(3.5)^{x}\).

Step2: Find initial value

The initial value of an exponential function of the form \(y = a(b)^{x}\) is the value when \(x = 0\). For \(g(x)=-2(3.5)^{x}\), when \(x = 0\), we have \(g(0)=-2(3.5)^{0}\). Since any non - zero number to the power of \(0\) is \(1\), \(g(0)=-2\times1=-2\). But in the given options, there was a wrong selection before. The correct initial value calculation: \(g(0)=-2\times(3.5)^{0}=-2\). Wait, maybe the user made a mistake in the original selection. Let's re - calculate:
For \(g(x)=-2(3.5)^{x}\), initial value (when \(x = 0\)): \(g(0)=-2\times(3.5)^{0}=-2\). But looking at the options, maybe the first part:
For the function definition: When reflecting \(f(x)=2(3.5)^{x}\) over \(x\) - axis, \(g(x)=-f(x)=-2(3.5)^{x}\), so the coefficient is \(-2\).
For initial value (\(x = 0\)): \(g(0)=-2\times(3.5)^{0}=-2\). But in the original wrong selection, it was \(-3.5\), which is incorrect. The correct initial value is \(-2\).
For \(g(1)\): Substitute \(x = 1\) into \(g(x)=-2(3.5)^{x}\), \(g(1)=-2\times3.5=-7\), which matches the given correct selection for \(g(1)\).
For \(g(-1)\): Substitute \(x=-1\) into \(g(x)=-2(3.5)^{x}\), \(g(-1)=-2\times(3.5)^{-1}=-2\times\frac{1}{3.5}=-\frac{2}{3.5}\approx - 0.57\), which matches the given correct selection for \(g(-1)\).

Answer:

• Function definition of \(g(x)\): \(g(x)=-2(3.5)^{x}\)
• Initial value of \(g(x)\): \(-2\)
• \(g(-1)\approx - 0.57\) (matches the given)
• \(g(1)=-7\) (matches the given)

But focusing on the initial value correction: The correct initial value of \(g(x)\) is \(-2\) (since when \(x = 0\), \(g(0)=-2\times(3.5)^{0}=-2\)). The wrong selection in the original was \(-3.5\), the correct one from the options (if we consider the options given: \(-3.5\), \(-2\), \(0\), \(2\)) is \(-2\).