Find the domain of \( f(x) = \sqrt{x-3} \).
\( x \geq 3 \).
Functions & Graphs
10 exam-style questions on domain, range, composite, inverse and modulus functions. Tap Show answer for full working.
3 Easy4 Medium3 Hard
Given \( f(x)=2x+1 \) and \( g(x)=x^2 \), find \( fg(x) \) and \( gf(x) \).
\( fg(x)=2x^2+1 \), \( gf(x)=(2x+1)^2 \).
Find the inverse of \( f(x)=3x-2 \).
\( f^{-1}(x)=\dfrac{x+2}{3} \).
Sketch \( y=|x-2| \) and state the range.
V-shape, vertex at \( (2,0) \). Range \( y \geq 0 \).
Given \( f(x)=e^x \) and \( g(x)=\ln x \), find \( fg(x) \) and \( gf(x) \) and state the domains.
\( fg(x)=x \) (domain \( x>0 \)); \( gf(x)=x \) (domain all real \( x \)).
Find the range of \( f(x)=x^2-4x+7 \) for \( x \in \mathbb{R} \).
Complete square: \( (x-2)^2+3 \). Range \( y \geq 3 \).
Solve \( |2x-1| = |x+3| \).
\( x=4 \) or \( x=-\dfrac{2}{3} \).
The function \( f \) is defined by \( f(x)=\dfrac{2x+1}{x-3} \), \( x eq 3 \). Find \( f^{-1}(x) \) and state its domain.
\( y=\dfrac{2x+1}{x-3} \) ⇒ \( y(x-3)=2x+1 \) ⇒ \( yx-2x=1+3y \) ⇒ \( x=\dfrac{1+3y}{y-2} \). So \( f^{-1}(x)=\dfrac{1+3x}{x-2} \), domain \( x eq 2 \).
Given \( f(x)=|x-1| \) and \( g(x)=2-x \), sketch \( y=fg(x) \) and find the set of values of \( x \) for which \( fg(x)=2 \).
\( fg(x)=|2-x-1|=|1-x| \). \( |1-x|=2 \) ⇒ \( x=-1 \) or \( x=3 \).
The function \( h \) is defined by \( h(x)=x^2-4 \), \( x\geq 0 \). Find \( h^{-1}(x) \) and sketch \( y=h(x) \) and \( y=h^{-1}(x) \) on the same axes.
\( y=x^2-4 \) ⇒ \( x=\sqrt{y+4} \) (since \( x\geq 0 \)). \( h^{-1}(x)=\sqrt{x+4} \), domain \( x\geq -4 \). Graphs are reflections in \( y=x \).