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Math  /  Algebra

Question2. Given that g(x)=12x+9g(x)=\frac{1}{2} x+9 and h(x)=(x+1)(x−1)h(x)=(x+1)(x-1), determine the following. Simplify where possible. a. h(x)−g(x)h(x)-g(x) [1 mark] b. h×g(x)h \times g(x) [2 marks] c. hg(x)\frac{h}{g}(x) [1 mark] d. g∘h(x)g \circ h(x) [2 marks] e. h∘g(x)h \circ g(x) [2 marks]

Studdy Solution

STEP 1

1. We are given two functions g(x)=12x+9 g(x) = \frac{1}{2}x + 9 and h(x)=(x+1)(x−1) h(x) = (x+1)(x-1) .
2. We need to perform operations on these functions as specified in parts a through e.
3. Simplification of expressions is required where possible.

STEP 2

1. Calculate h(x)−g(x) h(x) - g(x) .
2. Calculate h×g(x) h \times g(x) .
3. Calculate hg(x) \frac{h}{g}(x) .
4. Calculate the composition g∘h(x) g \circ h(x) .
5. Calculate the composition h∘g(x) h \circ g(x) .

STEP 3

Calculate h(x)−g(x) h(x) - g(x) .
First, find h(x) h(x) : h(x)=(x+1)(x−1)=x2−1 h(x) = (x+1)(x-1) = x^2 - 1
Now subtract g(x) g(x) : h(x)−g(x)=(x2−1)−(12x+9) h(x) - g(x) = (x^2 - 1) - \left(\frac{1}{2}x + 9\right)
Simplify the expression: h(x)−g(x)=x2−12x−10 h(x) - g(x) = x^2 - \frac{1}{2}x - 10

STEP 4

Calculate h×g(x) h \times g(x) .
Multiply h(x) h(x) and g(x) g(x) : h(x)×g(x)=(x2−1)×(12x+9) h(x) \times g(x) = (x^2 - 1) \times \left(\frac{1}{2}x + 9\right)
Distribute: =x2×12x+x2×9−1×12x−1×9 = x^2 \times \frac{1}{2}x + x^2 \times 9 - 1 \times \frac{1}{2}x - 1 \times 9
=12x3+9x2−12x−9 = \frac{1}{2}x^3 + 9x^2 - \frac{1}{2}x - 9

STEP 5

Calculate hg(x) \frac{h}{g}(x) .
Divide h(x) h(x) by g(x) g(x) : h(x)g(x)=x2−112x+9 \frac{h(x)}{g(x)} = \frac{x^2 - 1}{\frac{1}{2}x + 9}
This expression is already simplified as much as possible.

STEP 6

Calculate the composition g∘h(x) g \circ h(x) .
Substitute h(x) h(x) into g(x) g(x) : g(h(x))=g(x2−1) g(h(x)) = g(x^2 - 1)
=12(x2−1)+9 = \frac{1}{2}(x^2 - 1) + 9
Simplify: =12x2−12+9 = \frac{1}{2}x^2 - \frac{1}{2} + 9
=12x2+172 = \frac{1}{2}x^2 + \frac{17}{2}

STEP 7

Calculate the composition h∘g(x) h \circ g(x) .
Substitute g(x) g(x) into h(x) h(x) : h(g(x))=h(12x+9) h(g(x)) = h\left(\frac{1}{2}x + 9\right)
=(12x+9+1)(12x+9−1) = \left(\frac{1}{2}x + 9 + 1\right)\left(\frac{1}{2}x + 9 - 1\right)
Simplify: =(12x+10)(12x+8) = \left(\frac{1}{2}x + 10\right)\left(\frac{1}{2}x + 8\right)
Expand: =(12x)2+(12x)(8)+10(12x)+80 = \left(\frac{1}{2}x\right)^2 + \left(\frac{1}{2}x\right)(8) + 10\left(\frac{1}{2}x\right) + 80
=14x2+4x+5x+80 = \frac{1}{4}x^2 + 4x + 5x + 80
=14x2+9x+80 = \frac{1}{4}x^2 + 9x + 80

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