Algebra - Combining Functions (Assignment Problems)

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Section 3.6 : Combining Functions

Given \(f\left( x \right) = x + 12\) and \(g\left( x \right) = 9 + 4x\) compute each of the following.
1. \(f + g\)
2. \(\left( {f - g} \right)\left( 1 \right)\)
3. \(\left( {f\,g} \right)\left( x \right)\)
4. \(\displaystyle \frac{f}{g}\)
Given \(h\left( w \right) = {w^2} - 4w\) and \(f\left( w \right) = 2 + {w^2}\) compute each of the following.
1. \(\left( {h - f} \right)\left( w \right)\)
2. \(\left( {f + h} \right)\left( { - 4} \right)\)
3. \(f\,h\)
4. \(\displaystyle \left( {\frac{h}{f}} \right)\left( w \right)\)
Given \(A\left( x \right) = 6x - 1\) and \(\displaystyle P\left( x \right) = \frac{1}{{4 - x}}\) compute each of the following.
1. \(\left( {A + P} \right)\left( 0 \right)\)
2. \(\left( {P - A} \right)\left( { - 2} \right)\)
3. \(AP\)
4. \(\displaystyle \left( {\frac{A}{P}} \right)\left( x \right)\)
Given \(f\left( t \right) = 2t + 9\) and \(g\left( t \right) = 2t - 1\) compute each of the following.
1. \(\left( {f\,g} \right)\left( t \right)\)
2. \(\left( {f \circ g} \right)\left( t \right)\)
3. \(\left( {g \circ f} \right)\left( t \right)\)
4. \(\left( {g \circ g} \right)\left( t \right)\)
Given \(h\left( x \right) = {x^2} + 1\) and \(g\left( x \right) = 6 - 4x\) compute each of the following.
1. \(\left( {g\,h} \right)\left( x \right)\)
2. \(\left( {g \circ h} \right)\left( x \right)\)
3. \(\left( {h \circ g} \right)\left( x \right)\)
4. \(\left( {h \circ h} \right)\left( x \right)\)
Given \(A\left( w \right) = 2{w^2} + 9\) and \(R\left( w \right) = 1 - 2w - {w^2}\) compute each of the following.
1. \(\left( {A\,R} \right)\left( w \right)\)
2. \(\left( {A \circ R} \right)\left( w \right)\)
3. \(\left( {R \circ A} \right)\left( w \right)\)
4. \(\left( {A \circ A} \right)\left( w \right)\)
Given \(f\left( x \right) = 9{x^2} + 10x + 12\) and \(g\left( x \right) = 2\) compute each of the following.
1. \(\left( {f\,g} \right)\left( x \right)\)
2. \(\left( {g \circ f} \right)\left( x \right)\)
3. \(\left( {f \circ g} \right)\left( x \right)\)
4. \(\left( {g \circ g} \right)\left( x \right)\)
Given \(g\left( t \right) = t + 1\) and \(\displaystyle h\left( t \right) = \frac{2}{{t - 3}}\) compute each of the following.
1. \(\left( {g\,h} \right)\left( t \right)\)
2. \(\left( {g \circ h} \right)\left( t \right)\)
3. \(\left( {h \circ g} \right)\left( t \right)\)
4. \(\left( {h \circ h} \right)\left( x \right)\)
Given \(\displaystyle f\left( x \right) = \frac{1}{2}x - 3\) and \(g\left( x \right) = 2x + 6\), \(t \ge 0\) compute each of the following.
1. \(\left( {f \circ g} \right)\left( x \right)\)
2. \(\left( {g \circ f} \right)\left( x \right)\)
Given \(\displaystyle h\left( w \right) = \frac{1}{{w - 3}}\) and \(\displaystyle f\left( w \right) = \frac{{1 + 3w}}{w}\) compute each of the following.
1. \(\left( {h \circ f} \right)\left( w \right)\)
2. \(\left( {f \circ h} \right)\left( w \right)\)