Section 3.6 : Combining Functions
1. Given \(f\left( x \right) = 6x + 2\) and \(g\left( x \right) = 10 - 7x\) compute each of the following.
- \(\left( {f - g} \right)\left( 2 \right)\)
- \(\left( {g - f} \right)\left( 2 \right)\)
- \(f\,g\)
- \(\displaystyle \left( {\frac{f}{g}} \right)\left( x \right)\)
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a \(\left( {f - g} \right)\left( 2 \right)\) Show SolutionNot much to do here other than do the evaluation. We’ll leave it to you to verify the specific function evaluations.
\[\left( {f - g} \right)\left( 2 \right) = f\left( 2 \right) - g\left( 2 \right) = 14 - \left( { - 4} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{18}}\]b \(\left( {g - f} \right)\left( 2 \right)\) Show Solution
Not much to do here other than do the evaluation. We’ll leave it to you to verify the specific function evaluations.
\[\left( {g - f} \right)\left( 2 \right) = g\left( 2 \right) - f\left( 2 \right) = - 4 - 14 = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 18}}\]c \(f\,g\) Show Solution
Not much to do here other than do the evaluation. Also, remember that in this case the “evaluation” really only consists of plugging the two equations for each function in and doing some basic algebraic manipulations to simplify the answer if possible.
\[f\,g = f\left( x \right)g\left( x \right) = \left( {6x + 2} \right)\left( {10 - 7x} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 42{x^2} + 46x + 20}}\]d \(\displaystyle \left( {\frac{f}{g}} \right)\left( x \right)\) Show Solution
Not much to do here other than do the evaluation. Also, remember that in this case the “evaluation” really only consists of plugging the two equations for each function in and doing some basic algebraic manipulations to simplify the answer if possible, although in this case there really isn’t any simplification that can be done.
\[\left( {\frac{f}{g}} \right)\left( x \right) = \frac{{f\left( x \right)}}{{g\left( x \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{6x + 2}}{{10 - 7x}}}}\]