Math  /  Algebra

Question2. Given that g(x)=12x+9g(x)=\frac{1}{2} x+9 and h(x)=(x+1)(x1)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. gh(x)g \circ h(x) [2 marks] e. hg(x)h \circ g(x) [2 marks]

Studdy Solution
Calculate the composition hg(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+91) = \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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