2.2 Functions, domain, range and inverseIB Maths: Analysis and Approaches HL: Flashcards

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What makes a relation a function?

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What makes a relation a function?: Each input in the domain gives exactly one output.
What is the domain of a function?: The set of input values for which the function is defined.
What is the range of a function?: The set of output values the function actually takes.
State the largest possible domain of $f(x) = \sqrt{x+5}$ .: $x \ge -5$
State the range of $f(x) = \sqrt{x} - 4$ , $x \ge 0$ .: $f(x) \ge -4$
If $C(n) = 50 + 8n$ , what does $C(10)$ mean and what is it?: The output when $n = 10$ : $C(10) = 130$ .
Solving $f(x) = 10$ is equivalent to finding what?: $f^{-1}(10)$
If $f(3) = 11$ , what is $f^{-1}(11)$ ?: $3$
How is the graph of $y = f^{-1}(x)$ related to the graph of $y = f(x)$ ?: It is the reflection in the line $y = x$ .
How are the domain and range of $f^{-1}$ related to those of $f$ ?: Domain of $f^{-1}$ = range of $f$ ; range of $f^{-1}$ = domain of $f$ .
Which functions have an inverse function?: One-to-one functions: each output comes from only one input.
Why does $h(x) = x^2$ , $x \in \mathbb{R}$ , have no inverse?: It is not one-to-one: e.g. $h(-3) = h(3) = 9$ .
Find $f^{-1}(x)$ for $f(x) = 4x - 7$ .: $f^{-1}(x) = \frac{x + 7}{4}$