Describe the end behavior of the given function. \( f(x)=\left(\frac{2}{3}\right)^{z}-2 \) A. As \( x \) decreases, \( f(x) \) approaches 2 . B. As \( x \) decreases, \( f(x) \) approaches negative infinity. C. As \( x \) increases, \( f(x) \) approaches -2 . D. As \( x \) increases, \( f(x) \) approaches infinity.

To analyze the end behavior of the function \( f(z) = \left(\frac{2}{3}\right)^{z} - 2 \), we need to consider the limits of \( f(z) \) as \( z \) approaches positive and negative infinity. 1. **As \( z \) approaches positive infinity (\( z \to +\infty \)):** - The term \( \left(\frac{2}{3}\right)^{z} \) will approach 0 because \( \frac{2}{3} < 1 \). Therefore: \[ f(z) = \left(\frac{2}{3}\right)^{z} - 2 \to 0 - 2 = -2 \] 2. **As \( z \) approaches negative infinity (\( z \to -\infty \)):** - The term \( \left(\frac{2}{3}\right)^{z} \) will approach infinity because raising a fraction less than 1 to a negative exponent results in a large positive number. Therefore: \[ f(z) = \left(\frac{2}{3}\right)^{z} - 2 \to +\infty - 2 = +\infty \] Now, summarizing the end behavior: - As \( z \to +\infty \), \( f(z) \to -2 \). - As \( z \to -\infty \), \( f(z) \to +\infty \). Based on this analysis, the correct options are: - C. As \( z \) increases, \( f(z) \) approaches -2. - B. As \( z \) decreases, \( f(z) \) approaches positive infinity. Thus, the correct answers are C and B.