Question: The image contains a mathematical problem asking to identify which of the following four graphs is…

The image contains a mathematical problem asking to identify which of the following four graphs is the correct graph of the function \( g(x) = \left( \frac{2}{5} \right)^{-x} \).

The four graphs are shown below:

1. The first graph shows a curve that starts at the origin and increases as \( x \) increases.
2. The second graph shows a curve that starts at the origin and decreases as \( x \) increases.
3. The third graph shows a curve that starts at the origin and increases as \( x \) increases.
4. The fourth graph shows a curve that starts at the origin and decreases as \( x \) increases.

The function \( g(x) = \left( \frac{2}{5} \right)^{-x} \) is an exponential function with a base of \( \frac{2}{5} \) raised to the power of \( -x \).

Solution

To identify which of the following four graphs represents the function \( g(x) = \left(\frac{2}{5}\right)^{-x} \), we need to analyze its behavior. 1. The function can be rewritten using the negative exponent rule: \[ g(x) = \left(\frac{5}{2}\right)^{x} \] 2. The expression \(\left(\frac{5}{2}\right)^{x}\) indicates exponential growth because the base \(\frac{5}{2} > 1\). 3. The graph of \(g(x)\) should display the following characteristics: - It increases as \(x\) increases. - It passes through \( (0,1) \) because any base raised to the zero power equals 1. Based on these characteristics, the graph in the top left corner of the provided image, which shows increasing behavior, fits the function \(g(x) = \left(\frac{5}{2}\right)^{x}\).