A function of the form \( f(x)=a b^{x} \) is modified so that the \( b \) value remains the same but the a value is increased by 2 . How do the domain and range of the new function compare to the domain and range of the original function? Check all that apply. The range stays the same. The range becomes \( y>2 \). The domain stays the same. The domain becomes \( x>2 \). The range becomes \( y \geq 2 \). The domain becomes \( x \geq 2 \).

When modifying the function \( f(x) = a b^x \) to \( f_{\text{new}}(x) = (a + 2) b^x \), let's analyze how the domain and range are affected: 1. **Domain:** - **Original Function:** The domain of \( f(x) = a b^x \) is all real numbers, \( \mathbb{R} \), since exponential functions are defined for every real \( x \). - **Modified Function:** The new function \( f_{\text{new}}(x) = (a + 2) b^x \) retains the same exponential structure. Therefore, the domain remains all real numbers, \( \mathbb{R} \). 2. **Range:** - **Original Function:** Assuming \( a > 0 \) and \( b > 0 \) with \( b \neq 1 \), the range of \( f(x) = a b^x \) is \( y > 0 \). - **Modified Function:** By increasing \( a \) by 2, resulting in \( a + 2 \), the coefficient remains positive if \( a > -2 \) (which is a reasonable assumption to maintain the exponential behavior). Hence, the range of \( f_{\text{new}}(x) = (a + 2) b^x \) also remains \( y > 0 \). **Summary:** - **The domain stays the same.** - **The range stays the same.** **Checked Statements:** - The range stays the same. - The domain stays the same. Answer: All statements that “The range stays the same” and “The domain stays the same” are correct.