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 dive into the function \( g(x) \) and unravel its mysteries! First up, the **y-intercept**: To find it, we evaluate \( g(0) \) since that's where the function crosses the y-axis. From the definition of \( g(x) \), when \( x \geq 0 \), we have \( g(0) = -0^2 + 2 = 2 \). So yes, the y-intercept is indeed 2! As for the **continuity** of function \( g \): We check the limit from the left as \( x \) approaches 0, which gives us \( g(0^-) = \left(\frac{1}{2}\right)^{0} + 3 = 4 \). From the right, we have \( g(0^+) = 2 \). Since these two values don't match, the function is not continuous at \( x = 0 \). Now, about the **decreasing nature** of \( g(x) \): For \( x < 0 \), \( g(x) = \left(\frac{1}{2}\right)^x + 3 \), which is decreasing as it’s an exponential function (specifically with a base less than 1). However, for \( x \geq 0 \), \( g(x) = -x^2 + 2 \) is actually decreasing after reaching its maximum point at \( x = 0 \). So, the overall statement is somewhat true, but slightly misleading since the function increases on \( (-\infty, 0) \) and is decreasing afterwards. Regarding the statement about the function approaching infinity: As \( x \) approaches positive infinity, \( g(x) \) behaves like \( -x^2 + 2 \), which actually approaches negative infinity, not positive. Hence, this statement is false. Finally, oh yes, the **domain**: The function is defined for all real numbers—no gaps here! So, the domain is indeed all real numbers. In summary, the only true statements are about the y-intercept being 2 and the domain being all real numbers. They’re the shining stars in this function's analysis!