Flashcards · 10 cards

Functions

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Rule
What is $f \circ g(x)$ ?
Answer
$f\big(g(x)\big)$ : $g$ acts first. $g \circ f$ is usually different.
Order mattersf∘g: g first. g∘f: f first
From the lesson: Composite functions and inverses
Know it
Is $f^{-1}(x)$ the same as $\dfrac{1}{f(x)}$ ?
Answer
No. $f^{-1}$ undoes $f$ ; the $-1$ doesn't mean "one over".
From the lesson: Composite functions and inverses
Know it
How do you find $f^{-1}$ ?
Answer
Write $y = f(x)$ and make $x$ the subject. Then write the result as a function of $x$ .
From the lesson: Composite functions and inverses
Rule
How are the graphs of $f$ and $f^{-1}$ related?
Answer
They are reflections of each other in the line $y = x$ : $(3, 4)$ on $f$ gives $(4, 3)$ on $f^{-1}$ .
f and f⁻¹Reflections in y = x: (3, 4) on f, (4, 3) on f⁻¹
From the lesson: Composite functions and inverses
Rule
What makes a mapping a function?
Answer
Each input has exactly one output.
From domain to rangeEach input has exactly one output
From the lesson: Domain and range
Know it
Which values must be left out of the domain of $\dfrac{1}{x + 2}$ , and of $\sqrt{x - 3}$ ?
Answer
$x = -2$ , which would divide by zero; and $x < 3$ , which would take the square root of a negative number.
From the lesson: Domain and range
Rule
How do you find the range of a function?
Answer
Find the highest and lowest outputs and whether each is reached. For example $0 < y \le 4$ : 4 is reached, 0 never is.
Reading a rangeHighest value 4; never reaches 0: 0 < y ≤ 4
From the lesson: Domain and range
Know it
What is the domain of $f^{-1}$ ?
Answer
The range of $f$ .
From the lesson: Domain and range
Which method?
WAEC 2011 · Paper 2 · Q9 (b)
Given that $f: x \to x^2 + 1$ and $g: x \to \dfrac{3x + 1}{x - 1}$ , $x \ne 1$ , find: (i) $g\circ f$ ; (ii) $g\circ f(2)$ .
In $g \circ f$ , which function goes first?
Answer
$f$ . Work out $f(x) = x^2 + 1$ , then put that whole expression into $g$ wherever $x$ appears.
From the lesson: Composite functions and inverses Try the question
Which method?
WAEC 2022 · Paper 2 · Q3
Two functions $f$ and $g$ are defined on the set of real numbers, $\mathbb{R}$ , by $f : x \to x^2 + 2$ and $g : x \to \dfrac{1}{x + 2}$ , $x \ne -2$ . Find the domain of $(g \circ f)^{-1}$ .
How do you find the domain of $(g \circ f)^{-1}$ ?
Answer
It is the range of $g \circ f$ . Here $g \circ f(x) = \dfrac{1}{x^2 + 4}$ , which is largest, $\frac14$ , at $x = 0$ and never reaches 0.
From the lesson: Domain and range Try the question