domain and range \( h(x)=\left\{\begin{array}{ll}x^{2}, & -3 \leq x<3 \\ 2-x, & 3 \leq x<7\end{array}\right. \)

To determine the **domain** and **range** of the piecewise function: \[ h(x) = \begin{cases} x^{2}, & -3 \leq x < 3 \\ 2 - x, & 3 \leq x < 7 \end{cases} \] ### **Domain** The **domain** consists of all the \( x \)-values for which the function is defined. - **First piece**: \( x^2 \) is defined for \( -3 \leq x < 3 \). - **Second piece**: \( 2 - x \) is defined for \( 3 \leq x < 7 \). Combining these intervals, the domain of \( h(x) \) is: \[ \boxed{[-3,\ 7)} \] ### **Range** The **range** consists of all the possible \( h(x) \)-values based on the defined \( x \)-values. 1. **For \( -3 \leq x < 3 \), \( h(x) = x^2 \):** - The smallest value is \( 0 \) (when \( x = 0 \)). - As \( x \) approaches \( 3 \) (from below), \( h(x) \) approaches \( 9 \) but does not include \( 9 \). - **Range for this piece**: \( [0, 9) \). 2. **For \( 3 \leq x < 7 \), \( h(x) = 2 - x \):** - At \( x = 3 \), \( h(x) = 2 - 3 = -1 \). - As \( x \) approaches \( 7 \) (from below), \( h(x) \) approaches \( -5 \) but does not include \( -5 \). - **Range for this piece**: \( (-5, -1] \). 3. **Combining both ranges:** - The first piece covers \( [0, 9) \). - The second piece covers \( (-5, -1] \). - There is a gap between \( -1 \) and \( 0 \), meaning values in \( (-1, 0) \) are **not** included in the range. Thus, the **combined range** of \( h(x) \) is: \[ \boxed{\,(-5,\ -1\,] \ \cup\ [0,\ 9)\,} \] ### **Summary** - **Domain**: \( [-3,\ 7) \) - **Range**: \( (-5, -1] \cup [0, 9) \)