Question: Let \( f(x) = x^2 + 3 \) and \( h(x) = x + 7 \). Find the value or expression. \[ (f \circ…

Let \( f(x) = x^2 + 3 \) and \( h(x) = x + 7 \). Find the value or expression.

\[ (f \circ h)\left(\frac{1}{2}\right) \]

\[ (f \circ h)\left(\frac{1}{2}\right) = \boxed{\phantom{expression}} \]

(Simplify your answer.)

Solution

The problem asks us to find \((f \circ h)\left(\frac{1}{2}\right)\) where \(f(x) = x^2 + 3\) and \(h(x) = x + 7\). First, we find \(h\left(\frac{1}{2}\right)\). \[ h\left(\frac{1}{2}\right) = \frac{1}{2} + 7 = \frac{1}{2} + \frac{14}{2} \] \[ h\left(\frac{1}{2}\right) = \frac{15}{2} \] Next, we substitute this into \(f(x)\). \[ f\left(h\left(\frac{1}{2}\right)\right) = f\left(\frac{15}{2}\right) \] \[ f\left(\frac{15}{2}\right) = \left(\frac{15}{2}\right)^2 + 3 \] Calculate \(\left(\frac{15}{2}\right)^2\). \[ \left(\frac{15}{2}\right)^2 = \frac{225}{4} \] Now add 3 to \(\frac{225}{4}\). Convert 3 to a fraction with a denominator of 4: \[ 3 = \frac{12}{4} \] Add the fractions: \[ f\left(\frac{15}{2}\right) = \frac{225}{4} + \frac{12}{4} = \frac{237}{4} \] The final answer is \(\frac{237}{4}\).