Consider function \( g \). \( g(x)=\left\{\begin{array}{ll}\left(\frac{1}{2}\right)^{x}+3, & x<0 \\ -x^{2}+2, & x \geq 0\end{array}\right. \) Which statements are true about function \( g \) ? The \( y \)-intercept is 2 . The function is decreasing over its domain except for when \( x=0 \). The function is continuous. As \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity. The domain is all real numbers.

Let's analyze each statement about the function \( g(x) \): \[ g(x) = \begin{cases} \left(\frac{1}{2}\right)^x + 3 & \text{if } x < 0, \\ -x^2 + 2 & \text{if } x \geq 0. \end{cases} \] 1. **The \( y \)-intercept is 2.** - **True.** The \( y \)-intercept occurs at \( x = 0 \). For \( x \geq 0 \): \[ g(0) = -0^2 + 2 = 2. \] 2. **The function is decreasing over its domain except for when \( x = 0 \).** - **True.** - For \( x < 0 \): \[ g(x) = \left(\frac{1}{2}\right)^x + 3. \] The term \( \left(\frac{1}{2}\right)^x = 2^{-x} \) is a decreasing function because the base \( \frac{1}{2} < 1 \). Adding 3 shifts it upward but does not affect the decreasing nature. - For \( x \geq 0 \): \[ g(x) = -x^2 + 2. \] This is a downward-opening parabola, which is decreasing for \( x > 0 \). - At \( x = 0 \), there is a jump discontinuity, but the function remains decreasing on either side of \( x = 0 \). 3. **The function is continuous.** - **False.** There is a discontinuity at \( x = 0 \): - Left-hand limit as \( x \to 0^- \): \[ \lim_{x \to 0^-} g(x) = \left(\frac{1}{2}\right)^0 + 3 = 1 + 3 = 4. \] - Right-hand value at \( x = 0 \): \[ g(0) = 2. \] - Since \( 4 \neq 2 \), \( g(x) \) is not continuous at \( x = 0 \). 4. **As \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity.** - **False.** For \( x \geq 0 \): \[ \lim_{x \to +\infty} g(x) = \lim_{x \to +\infty} (-x^2 + 2) = -\infty. \] 5. **The domain is all real numbers.** - **True.** The function \( g(x) \) is defined for all \( x < 0 \) and \( x \geq 0 \), covering the entire set of real numbers. **Summary:** - **True Statements:** 1. The \( y \)-intercept is 2. 2. The function is decreasing over its domain except for when \( x = 0 \). 5. The domain is all real numbers. - **False Statements:** 3. The function is continuous. 4. As \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity. **Final Answer:** All statements except the third and fourth are true. In other words, statements 1, 2 and 5 are true.