## Intermediate Algebra: Connecting Concepts through Application

a) $h(32)=-9$ b) $x =15$ c) $\text{Domain: all real numbers} \\\text{Range: all real numbers}$
a) To find $h(32),$ substitute $x$ with $32$ in the given function, $h(x)=-\dfrac{5}{8}x+11.$ Hence, \begin{array}{l}\require{cancel} h(x)=-\dfrac{5}{8}x+11 \\\\ h(32)=-\dfrac{5}{8}(32)+11 \\\\ h(32)=-20+11 \\\\ h(32)=-9 .\end{array} b) Replace $h(x)$ with $\dfrac{13}{8}$ in the given function, $h(x)=-\dfrac{5}{8}x+11.$ Hence, \begin{array}{l}\require{cancel} h(x)=-\dfrac{5}{8}x+11 \\\\ \dfrac{13}{8}=-\dfrac{5}{8}x+11 \\\\ 8\left( \dfrac{13}{8} \right) =8\left( -\dfrac{5}{8}x+11 \right) \\\\ 13 =-5x+88 \\\\ 5x =88-13 \\\\ 5x =75 \\\\ x =\dfrac{75}{5} \\\\ x =15 .\end{array} c) The domain and the range of linear functions in the form $f(x)=mx+b$ is the set of all real numbers. Hence, the given function has the following characteristics: \begin{array}{l}\require{cancel} \text{Domain: all real numbers} \\\text{Range: all real numbers} .\end{array}