Study Mathematics notes on Functions and Quadratics for Secondary. Useful for revision, homework, and exam preparation in Zimbabwe.
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1(a) Find $(\sigma \cdot g)(x)$.
Given:
\[
f(x)=x^{2}+4 \quad \text { and } \quad g(x)=x-1
\]
The composition $(\sigma \cdot g)(x)$ is not standard notation. Assuming it means $(f \circ g)(x)$ (i.e., $f(g(x))$ ):
\[
(f \circ g)(x)=f(g(x))=f(x-1)=(x-1)^{2}+4=x^{2}-2 x+1+4=x^{2}-2 x+5
\]
Answer: $x^{2}-2 x+5$
1(b) Find the coordinates of the vertex of the graph of $h$.
The graph of $f \circ g$ is translated by the vector $\binom{3}{-1}$ to obtain $h(x)$. First, express $h(x)$ :
\[
h(x)=(f \circ g)(x-3)-1=(x-3)^{2}-2(x-3)+5-1=x^{2}-6 x+9-2 x+6+4=x^{2}-8 x+19
\]
The vertex form of a quadratic $a x^{2}+b x+c$ has its vertex at $x=-\frac{b}{2 a}$ :
\[
x=-\frac{-8}{2 \times 1}=4
\]
Substitute $x=4$ into $h(x)$ :
\[
h(4)=4^{2}-8 \times 4+19=16-32+19=3
\]
Answer: The vertex is at $(4,3)$.
1(c) Show that $h(x)=x^{2}-8 x+19$.
This was derived in part (b):
\[
h(x)=x^{2}-8 x+19
\]
Answer: As shown above, $h(x)=x^{2}-8 x+19$.
1(d) Find the $x$-coordinate of point $P$ where the line $y=2 x-6$ is tangent to the graph of $h$. Set $h(x)=2 x-6$ :
\[
x^{2}-8 x+19=2 x-6 x^{2}-10 x+25=0
\]
Solve the quadratic equation:
\[
x=\frac{10 \pm \sqrt{100-100}}{2}=\frac{10}{2}=5
\]
Answer: The $x$-coordinate is 5 .
2(a) Find an expression for $(h \circ k)(x)$.
Given:
\[
g(x)=3 x-2 \quad \text { and } \quad h(x)=\frac{5 x}{x-4}, \quad x \neq 4
\]
Assuming $k(x)$ is a typo and should be $g(x)$ :
\[
(h \circ g)(x)=h(g(x))=h(3 x-2)=\frac{5(3 x-2)}{(3 x-2)-4}=\frac{15 x-10}{3 x-6}=\frac{5(3 x-2)}{3(x-2)}
\]
Simplify:
\[
(h \circ g)(x)=\frac{15 x-10}{3 x-6}=\frac{5(3 x-2)}{3(x-2)}
\]