IB›IB Maths: Analysis and Approaches SL›Mind maps2.2 Functions, domain, range and inverseIB Maths: Analysis and Approaches SL: Mind mapStudy pack PDFAlso for this subtopic:Revision notesFlashcardsSubtopic testCover factsWhat is a function?Exactly one output per inputDomain = inputs, range = outputsVertical line test: at most one hitx−3\sqrt{x-3}x−3 needs x≥3x \ge 3x≥3; 1x+2\frac{1}{x+2}x+21 needs x≠−2x \ne -2x=−2Notation, modelsC(12)C(12)C(12) means substitute 12, not C×12C\times12C×12Model domain must make sense in contextTaxi fare C(d)=3.5+1.8dC(d) = 3.5 + 1.8dC(d)=3.5+1.8dRange from values at the ends of the domainInverse functionf(a)=b⇔f−1(b)=af(a) = b \Leftrightarrow f^{-1}(b) = af(a)=b⇔f−1(b)=aSolving f(x)=10f(x) = 10f(x)=10 is finding f−1(10)f^{-1}(10)f−1(10)Write y=f(x)y = f(x)y=f(x), swap xxx and yyy, rearrangeF−1(x)=x−321.8F^{-1}(x) = \frac{x-32}{1.8}F−1(x)=1.8x−32Functionsdomain, range, inversef(x)f(x)f(x)f−1f^{-1}f−1Inverse graphReflection in the line y=xy = xy=x(a,b)(a,b)(a,b) becomes (b,a)(b,a)(b,a)Domain of f−1f^{-1}f−1 = range of fffRange of f−1f^{-1}f−1 = domain of fffWhen it existsFunction must be one-to-oneHorizontal line test: at most one hitTwo inputs, same output: not one-to-oneRestrict the domain to fix itExam tipsf−1(x)f^{-1}(x)f−1(x) is not 1f(x)\frac{1}{f(x)}f(x)1Reject solutions outside the restricted domainCheck: F(180)=356F(180) = 356F(180)=356, so F−1(356)=180F^{-1}(356) = 180F−1(356)=180Square root is always ≥0\ge 0≥0, helps with range