Question: Indicate which of the following four graphs is the correct graph of this function. \[ g(x) =…

Indicate which of the following four graphs is the correct graph of this function.

\[ g(x) = \left( \frac{2}{5} \right)^{3-x} \]

Answer

Keypad

Keyboard Shortcuts

Solution

To determine which graph is the correct one for the function \[ q(x) = \left( \frac{2}{5} \right)^{3-x} \] first analyze the function: Rewrite the function in the form of an exponential expression: \[ q(x) = \left( \frac{2}{5} \right)^3 \cdot \left( \frac{2}{5} \right)^{-x} \] Since \(\frac{2}{5} < 1\), the function \(\left( \frac{2}{5} \right)^{-x}\) is increasing because it is an exponential with a base between 0 and 1. This transformation will have: For \( x \to -\infty \), \( \left( \frac{2}{5} \right)^{-x} \to \infty \) For \( x \to \infty \), \( \left( \frac{2}{5} \right)^{-x} \to 0 \) This behavior matches a decreasing curve starting from high on the left side and approaching the x-axis (y = 0) asymptote from above as \( x \) increases. The top left graph, which shows a curve starting above and decreasing as it moves to the right, represents this behavior. Therefore, this is the correct graph for the function \( q(x) \).