Find function from composite - Composite & Inverse Functions

CAIE P1 2024 March Q9

9 marks Standard +0.3

9 The functions f and g are defined for all real values of $x$ by $$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$ where $k$ is a constant.

Given that the range of the function gf is $\mathrm { gf } ( x ) \geqslant 39$, find the value of $k$.
For this value of $k$, determine the range of the function fg .
The function h is defined for all real values of $x$ and is such that $\mathrm { gh } ( x ) = 35 x + 19$. Find an expression for $\mathrm { g } ^ { - 1 } ( x )$ and hence, or otherwise, find an expression for $\mathrm { h } ( x )$. \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721} The diagram shows the circle with centre $C ( - 4,5 )$ and radius $\sqrt { 20 }$ units. The circle intersects the $y$-axis at the points $A$ and $B$. The size of angle $A C B$ is $\theta$ radians.

CAIE P1 2010 November Q7

7 marks Standard +0.3

7 The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$

Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$ and hence state the range of f .
Obtain an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$. The function g is defined by $$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$ The function h is such that $\mathrm { f } = \mathrm { hg }$ and the domain of h is $x > 0$.
Obtain an expression for $\mathrm { h } ( x )$.

CAIE P1 2016 November Q8

8 marks Moderate -0.3

Express $4x^2 + 12x + 10$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are constants. [3]
Functions $f$ and $g$ are both defined for $x > 0$. It is given that $f(x) = x^2 + 1$ and $fg(x) = 4x^2 + 12x + 10$. Find $g(x)$. [1]
Find $(fg)^{-1}(x)$ and give the domain of $(fg)^{-1}$. [4]