UCTIONS: Answer any FIVE (5) Questions a) Find the domain and range of the function \( f(x)=\left\{\begin{array}{ll}x+1 & \text { if }-1

Ask by Mckenzie Gray. in Nigeria

Mar 10,2025

1. **Determine the Domain** - The function \( f(x) \) is given by: \[ f(x) = \begin{cases} x+1 & \text{if } -1 < x < 1, \\[6pt] 2 & \text{if } 1 \leq x. \end{cases} \] - The first piece is defined for \( x \) satisfying \( -1 < x < 1 \) and the second piece is defined for \( x \) satisfying \( 1 \leq x \). - Combining these intervals, the domain is: \[ (-1, 1) \cup [1, \infty) = (-1,\infty). \] 2. **Determine the Range** - For \( -1 < x < 1 \), the function is \( f(x) = x + 1 \). - As \( x \) approaches \( -1 \) (but never equals \(-1\)), \( f(x) \) approaches \(-1+1=0 \). However, \( f(x) \) never actually reaches \( 0 \). - As \( x \) approaches \( 1 \) (but \( 1 \) is not included in this interval), \( f(x) \) approaches \( 1+1=2 \). - Therefore, the values of \( f(x) = x+1 \) as \( x \) varies in \( (-1,1) \) lie in the interval: \[ (0,2). \] - For \( x \geq 1 \), \( f(x) = 2 \) is a constant value. - Combining these results, the range is: \[ (0,2) \cup \{2\} = (0,2]. \] 3. **Final Answer** - Domain: \( (-1,\infty) \) - Range: \( (0,2] \)