1. **Problem statement:** Given a function $f(x)$ with domain $[0,8]$ and range $[-2,5]$, find the domain and range of the following functions: (a) $g(x) = f(x+2) - 3$ (b) $h(x) = -f(x)$ (c) $k(x) = f(-x) + 4$ (d) $m(x) = 2f(3x)$ (e) A new function obtained by reflecting $f(x)$ over the x-axis, vertically stretching by 2, and shifting up by 1. --- 2. **Recall:** - The domain of $f(x+a)$ shifts the domain by $-a$. - The range changes by vertical shifts and stretches. - Reflection over the x-axis changes $y$ to $-y$. - Vertical stretch by factor $c$ multiplies the range by $c$. - Vertical shift adds a constant to the range. --- 3. **(a) Find domain and range of $g(x) = f(x+2) - 3$** - Domain of $g$: Solve $x+2 \\in [0,8]$ so $x \\in [-2,6]$. - Range of $g$: Since $g(x) = f(x+2) - 3$, subtract 3 from all $f$ values. - Original range: $[-2,5]$, new range: $[-2 - 3, 5 - 3] = [-5, 2]$. --- 4. **(b) Find domain and range of $h(x) = -f(x)$** - Domain of $h$: same as $f$, $[0,8]$. - Range of $h$: multiply original range by $-1$ and reverse order. - Original range: $[-2,5]$, new range: $[-5,2]$ after reflection. --- 5. **(c) Find domain and range of $k(x) = f(-x) + 4$** - Domain of $k$: Solve $-x \\in [0,8]$ so $x \\in [-8,0]$. - Range of $k$: add 4 to original range. - Original range: $[-2,5]$, new range: $[-2 + 4, 5 + 4] = [2,9]$. --- 6. **(d) Find domain and range of $m(x) = 2f(3x)$** - Domain of $m$: Solve $3x \\in [0,8]$ so $x \\in [0, \frac{8}{3}]$. - Range of $m$: multiply original range by 2. - Original range: $[-2,5]$, new range: $[2 \times (-2), 2 \times 5] = [-4,10]$. --- 7. **(e) New function after reflection over x-axis, vertical stretch by 2, and shift up 1:** - Reflection: $f(x) \to -f(x)$, range $[-5,2]$ from step (b). - Vertical stretch by 2: multiply range by 2. - Shift up 1: add 1 to range. Calculate stepwise: - After reflection: range $[-5,2]$ - After stretch: $2 \times [-5,2] = [-10,4]$ - After shift: $[-10 + 1, 4 + 1] = [-9,5]$ Domain remains $[0,8]$ as no horizontal changes. --- **Final answers:** - (a) Domain: $[-2,6]$, Range: $[-5,2]$ - (b) Domain: $[0,8]$, Range: $[-5,2]$ - (c) Domain: $[-8,0]$, Range: $[2,9]$ - (d) Domain: $[0, \frac{8}{3}]$, Range: $[-4,10]$ - (e) Domain: $[0,8]$, Range: $[-9,5]$