1. Given functions and operations: \( f(x) = \sqrt{x} + 5, \quad g(x) = \frac{x - 10}{3}, \quad h(x) = (x + 2)^2 \) **a) Find and simplify \((h \circ g)(x) = h(g(x))\):** 1. Substitute \(g(x)\) into \(h(x)\): $$ (h \circ g)(x) = h\left(\frac{x - 10}{3}\right) = \left(\frac{x - 10}{3} + 2\right)^2 $$ 2. Simplify inside the parenthesis: $$ \frac{x - 10}{3} + 2 = \frac{x - 10}{3} + \frac{6}{3} = \frac{x - 10 + 6}{3} = \frac{x - 4}{3} $$ 3. Square the result: $$ \left(\frac{x - 4}{3}\right)^2 = \frac{(x - 4)^2}{9} $$ Hence, $$ (h \circ g)(x) = \frac{(x - 4)^2}{9} $$ **b) Determine \((h \circ g)(22)\):** 1. Substitute \(x = 22\) into the simplified expression: $$ (h \circ g)(22) = \frac{(22 - 4)^2}{9} = \frac{18^2}{9} $$ 2. Calculate the numerator: $$ 18^2 = 324 $$ 3. Divide: $$ \frac{324}{9} = 36 $$ So, $$ (h \circ g)(22) = 36 $$ 2. Graph the transformed cube root function: $$ y = -2 \sqrt[3]{\frac{1}{4}(x - 4)} + 1 $$ - This function involves a cube root transformation of \( x \). - The inside transformation shifts the graph right by 4 units (because of \(x - 4\)). - The factor \(\frac{1}{4}\) inside compresses the graph horizontally by a factor of 4. - The coefficient \(-2\) outside vertically stretches the graph by a factor of 2 and reflects it over the x-axis. - Finally, adding 1 shifts the graph up by 1 unit. 3. Describe the transformations for $$ G(x) = -5 \sqrt{12 - 4x} + 2 $$ - The factor \(-5\) indicates a vertical stretch by 5 and reflection about the x-axis. - Inside the root, \(12 - 4x\) can be rewritten as \( -4x + 12\): - The \(-4\) means a horizontal compression by a factor of \(\frac{1}{4}\) and reflection about the y-axis. - The \(+12\) shifts the graph horizontally to the right by 3 units (since setting \(12 - 4x = 0\) gives \(x=3\), the starting point). - The \(+2\) outside shifts the graph vertically up by 2 units. Summary of transformations: - Vertical stretch by 5 and reflection over the x-axis. - Horizontal compression by 4 and reflection over the y-axis. - Shift right by 3 units. - Shift up by 2 units.