Answer on paper first, then reveal the mark scheme one mark at a time.
Let $f(x)=2x+3$ and $g(x)=x^2$.
(a) Find $(f\circ g)(2)$. [2]
(b) Find $f^{-1}(x)$. [2]
The function $f$ is defined by $f(x)=\sqrt{x-2}+1$ for $x\ge2$.
(a) Write down the range of $f$. [1]
(b) Find $f^{-1}(x)$. [3]
(c) Write down the domain of $f^{-1}$. [1]
The point $A(4,-2)$ lies on the graph of $y=f(x)$. Find the coordinates of the image of $A$ on the graph of
(a) $y=f(x-3)+1$ [1]
(b) $y=-2f(x)$ [1]
(c) $y=f(2x)$ [1]
(d) $y=3f(2x-4)$ [2]
Let $f(x)=2x^2-8x+5$.
(a) Write $f(x)$ in the form $a(x-h)^2+c$. [3]
(b) Write down the coordinates of the vertex of the graph of $f$. [1]
(c) The equation $f(x)=k$ has two distinct real roots. Find the possible values of $k$. [2]
Let $f(x)=\dfrac{2x+1}{x-3}$, $x\ne3$. The graph of $y=f(x)$ is shown, with its asymptotes drawn dashed. The graph crosses the y-axis at P and the x-axis at Q.
(a) Write down the equations of the vertical and horizontal asymptotes. [2]
(b) Find the coordinates of P and Q. [2]
(c) Find $f^{-1}(x)$ and state its domain. [3]
(a) The function $f$ is defined by $f(x)=3x+a$, where $a$ is a constant. Given that $f(f(x))=9x+8$, find the value of $a$. [3]
(b) The function $h$ is defined by $h(x)=\dfrac{x+1}{x-1}$, $x\ne1$. Show that $h^{-1}(x)=h(x)$. [3]
The graph of $y=x^2+(k-3)x+k$ touches the x-axis.
(a) Find the possible values of $k$. [4]
(b) For each value of $k$, find the coordinates of the point where the graph touches the x-axis. [2]
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