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Section 1.1 : Review : Functions
1. Perform the indicated function evaluations for \(f\left( x \right) = 3 - 5x - 2{x^2}\).
- \(f\left( 4 \right) \)
- \(f\left( 0 \right)\)
- \(f\left( { - 3} \right) \)
- \(f\left( {6 - t} \right) \)
- \(f\left( {7 - 4x} \right)\)
- \(f\left( {x + h} \right) \)
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a \(f\left( 4 \right) \) Show Solution
\[f\left( 4 \right) = 3 - 5\left( 4 \right) - 2{\left( 4 \right)^2} = - 49\]
b \(f\left( 0 \right)\) Show Solution
\[f\left( 0 \right) = 3 - 5\left( 0 \right) - 2{\left( 0 \right)^2} = 3\]
c \(f\left( { - 3} \right)\) Show Solution
\[f\left( { - 3} \right) = 3 - 5\left( { - 3} \right) - 2{\left( { - 3} \right)^2} = 0\]
Hint : Don’t let the fact that there are now variables here instead of numbers get you confused. This works exactly the same way as the first three it will just have a little more algebra involved.
\[\begin{align*}f\left( {6 - t} \right) & = 3 - 5\left( {6 - t} \right) - 2{\left( {6 - t} \right)^2}\\ & = 3 - 5\left( {6 - t} \right) - 2\left( {36 - 12t + {t^2}} \right)\\ & = 3 - 30 + 5t - 72 + 24t - 2{t^2}\\ & = - 99 + 29t - 2{t^2}\end{align*}\]
Hint : Don’t let the fact that there are now variables here instead of numbers get you confused. This works exactly the same way as the first three it will just have a little more algebra involved.
\[\begin{align*}f\left( {7 - 4x} \right) & = 3 - 5\left( {7 - 4x} \right) - 2{\left( {7 - 4x} \right)^2}\\ & = 3 - 5\left( {7 - 4x} \right) - 2\left( {49 - 56x + 16{x^2}} \right)\\ & = 3 - 35 + 20x - 98 + 112x - 32{x^2}\\ & = - 130 + 132x - 32{x^2}\end{align*}\]
Hint : Don’t let the fact that there are now variables here instead of numbers get you confused. Also, don’t get excited about the fact that there is both an \(x\) and an \(h\) here. This works exactly the same way as the first three it will just have a little more algebra involved.
\[\begin{align*}f\left( {x + h} \right) & = 3 - 5\left( {x + h} \right) - 2{\left( {x + h} \right)^2}\\ & = 3 - 5\left( {x + h} \right) - 2\left( {{x^2} + 2xh + {h^2}} \right)\\ & = 3 - 5x - 5h - 2{x^2} - 4xh - 2{h^2}\end{align*}\]