Question

h(x) = x² + 1 k(x) = x - 2
(h + k)(2) =
done
(h - k)(3) =
done
evaluate 3h(2) + 2k(3) = .
done

Explanation:

For \((h + k)(2)\):

Step1: Recall the definition of \((h + k)(x)\)

The sum of two functions \(h(x)\) and \(k(x)\) is defined as \((h + k)(x)=h(x)+k(x)\). So, \((h + k)(2)=h(2)+k(2)\).

Step2: Evaluate \(h(2)\)

Given \(h(x)=x^{2}+1\), substitute \(x = 2\) into \(h(x)\):
\(h(2)=2^{2}+1=4 + 1=5\)

Step3: Evaluate \(k(2)\)

Given \(k(x)=x - 2\), substitute \(x = 2\) into \(k(x)\):
\(k(2)=2-2 = 0\)

Step4: Calculate \((h + k)(2)\)

\((h + k)(2)=h(2)+k(2)=5 + 0=5\)

For \((h - k)(3)\):

Step1: Recall the definition of \((h - k)(x)\)

The difference of two functions \(h(x)\) and \(k(x)\) is defined as \((h - k)(x)=h(x)-k(x)\). So, \((h - k)(3)=h(3)-k(3)\).

Step2: Evaluate \(h(3)\)

Given \(h(x)=x^{2}+1\), substitute \(x = 3\) into \(h(x)\):
\(h(3)=3^{2}+1=9 + 1=10\)

Step3: Evaluate \(k(3)\)

Given \(k(x)=x - 2\), substitute \(x = 3\) into \(k(x)\):
\(k(3)=3-2=1\)

Step4: Calculate \((h - k)(3)\)

\((h - k)(3)=h(3)-k(3)=10-1 = 9\)

For \(3h(2)+2k(3)\):

Step1: Evaluate \(h(2)\) (already found earlier)

We know from before that \(h(2)=5\)

Step2: Evaluate \(k(3)\) (already found earlier)

We know from before that \(k(3)=1\)

Step3: Substitute into the expression

Substitute \(h(2) = 5\) and \(k(3)=1\) into \(3h(2)+2k(3)\):
\(3h(2)+2k(3)=3\times5+2\times1\)

Step4: Perform the arithmetic

First, calculate the multiplications: \(3\times5 = 15\) and \(2\times1=2\)
Then, add the results: \(15 + 2=17\)

Answer:

s:
\((h + k)(2)=\boldsymbol{5}\)

\((h - k)(3)=\boldsymbol{9}\)

\(3h(2)+2k(3)=\boldsymbol{17}\)