IB Maths AA · Topic 2 · 32 cards

Functions flashcards

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What is a function, and how does the vertical line test relate to it?
A function gives exactly one output for each input. A graph is a function if every vertical line crosses it at most once.
What are the domain and range of a function?
The domain is the set of all allowed inputs (\(x\)-values). The range is the set of all outputs (\(y\)-values) that actually occur.
How do you find a composite function \((f\circ g)(x)\), and does the order matter?
\((f\circ g)(x)=f(g(x))\): apply \(g\) first, then \(f\). The order generally matters: \(f(g(x))\neq g(f(x))\) in most cases.
Calculation: \(f(x)=2x+3\) and \(g(x)=x^2\). Find \((f\circ g)(2)\) and \((g\circ f)(2)\).
\(g(2)=4\), so \(f(g(2))=2(4)+3=11\). \(f(2)=7\), so \(g(f(2))=7^2=49\).
How do you find the inverse function \(f^{-1}(x)\)?
Write \(y=f(x)\), swap \(x\) and \(y\), then rearrange to make \(y\) the subject. The graph of \(f^{-1}\) is the reflection of \(f\) in the line \(y=x\).
How are the domain and range of \(f\) and \(f^{-1}\) related?
They swap: the domain of \(f^{-1}\) is the range of \(f\), and the range of \(f^{-1}\) is the domain of \(f\). Only one-to-one functions have an inverse.
What are the main forms of the equation of a straight line?
Gradient-intercept \(y=mx+c\); point-gradient \(y-y_1=m(x-x_1)\); general \(ax+by+d=0\). Gradient \(m=\frac{y_2-y_1}{x_2-x_1}\).
How are the gradients of parallel and perpendicular lines related?
Parallel lines have equal gradients: \(m_1=m_2\). Perpendicular lines satisfy \(m_1m_2=-1\), so \(m_2=-\frac1{m_1}\).
Calculation: a line passes through \((1,2)\) and \((5,10)\). Find its equation and the gradient of a perpendicular line.
\(m=\frac{10-2}{5-1}=2\), so \(y-2=2(x-1)\), giving \(y=2x\). A perpendicular line has gradient \(-\frac12\).
What are the three standard forms of a quadratic function and what does each show?
\(ax^2+bx+c\) shows the \(y\)-intercept \(c\); \(a(x-p)(x-q)\) shows the roots \(p,q\); \(a(x-h)^2+k\) shows the vertex \((h,k)\).
What does the discriminant \(\Delta=b^2-4ac\) tell you about \(ax^2+bx+c=0\)?
\(\Delta>0\): two distinct real roots. \(\Delta=0\): one repeated real root. \(\Delta<0\): no real roots (graph does not cross the \(x\)-axis).
What is the equation of the axis of symmetry of \(y=ax^2+bx+c\)?
\(x=-\frac{b}{2a}\). The vertex lies on this line, so its \(y\)-coordinate is found by substituting this \(x\)-value.
Calculation: find the vertex and roots of \(y=x^2-6x+5\).
Axis \(x=\frac{6}{2}=3\); \(y=9-18+5=-4\), so the vertex is \((3,-4)\). Factorising \((x-1)(x-5)\) gives roots \(x=1\) and \(x=5\).
How can you show that a line is tangent to a quadratic curve?
Set the two equations equal to get a quadratic in \(x\). The line is tangent exactly when this quadratic has \(\Delta=0\) (one repeated solution).
What are the asymptotes of \(f(x)=\frac{ax+b}{cx+d}\)?
Vertical asymptote \(x=-\frac dc\) (where the denominator is zero) and horizontal asymptote \(y=\frac ac\). The reciprocal \(y=\frac1x\) has asymptotes \(x=0\) and \(y=0\).
What are the key features of the graph of \(y=a^x\) for \(a>1\)?
It passes through \((0,1)\), is always positive, increases, and has the \(x\)-axis (\(y=0\)) as a horizontal asymptote. For \(0
How are \(y=\ln x\) and \(y=e^x\) related graphically, and what are the features of \(y=\ln x\)?
They are inverses, so each is the reflection of the other in \(y=x\). \(y=\ln x\) passes through \((1,0)\), has domain \(x>0\) and the \(y\)-axis as a vertical asymptote.
How does \(y=f(x-a)+b\) transform the graph of \(y=f(x)\)?
It translates the graph by the vector \(\begin{pmatrix}a\\b\end{pmatrix}\): \(a\) units right and \(b\) units up (negative values go left and down).
How do \(y=pf(x)\) and \(y=f(qx)\) stretch the graph of \(y=f(x)\)?
\(y=pf(x)\) is a vertical stretch, scale factor \(p\), from the \(x\)-axis. \(y=f(qx)\) is a horizontal stretch, scale factor \(\frac1q\), from the \(y\)-axis.
How do \(y=-f(x)\) and \(y=f(-x)\) transform the graph of \(y=f(x)\)?
\(y=-f(x)\) reflects it in the \(x\)-axis. \(y=f(-x)\) reflects it in the \(y\)-axis.
Calculation: \((4,6)\) is on \(y=f(x)\). Where does it map to on \(y=2f(x-1)+3\)?
\(x\) becomes \(4+1=5\). \(y\) becomes \(2(6)+3=15\). The image point is \((5,15)\).
Calculation: solve the inequality \(x^2-3x-4<0\).
\(x^2-3x-4=(x-4)(x+1)\), so the roots are \(-1\) and \(4\). The parabola opens upwards, so it is below the axis between the roots: \(-1
(HL) What does the factor theorem say?
\((x-a)\) is a factor of a polynomial \(P(x)\) if and only if \(P(a)=0\). So every real root gives a linear factor.
(HL) What does the remainder theorem say? Calculation: the remainder when \(x^3-2x+5\) is divided by \((x-2)\).
The remainder on dividing \(P(x)\) by \((x-a)\) is \(P(a)\). Here \(P(2)=8-4+5=9\).
(HL) What are the sum and product of the roots of a quadratic and of a cubic?
Quadratic \(ax^2+bx+c\): sum \(-\frac ba\), product \(\frac ca\). Cubic \(ax^3+bx^2+cx+d\): sum \(-\frac ba\), sum of pairs \(\frac ca\), product \(-\frac da\).
(HL) What does the fundamental theorem of algebra say?
A polynomial of degree \(n\) has exactly \(n\) roots over the complex numbers, counting repeats. If the coefficients are real, non-real roots occur in conjugate pairs.
(HL) How are the graphs of \(y=|f(x)|\) and \(y=f(|x|)\) obtained from \(y=f(x)\)?
\(y=|f(x)|\): reflect any part below the \(x\)-axis upwards. \(y=f(|x|)\): keep the part with \(x\ge0\) and reflect it in the \(y\)-axis (the \(x<0\) part is discarded).
(HL) How do you sketch \(y=\frac{1}{f(x)}\) from the graph of \(y=f(x)\)?
Zeros of \(f\) become vertical asymptotes. Where \(f\) is large, \(\frac1f\) is near zero. Sign is unchanged. Points where \(f=\pm1\) stay fixed. Maxima become minima and vice versa.
(HL) How is the graph of \(y=f(ax+b)\) obtained from \(y=f(x)\)? Take \(y=f(2x+6)\).
Write \(f(2(x+3))\): horizontal stretch scale factor \(\frac12\) from the \(y\)-axis, then translate 3 units left. Equivalently translate 6 left first, then stretch.
(HL) How do you solve \(|x-1|=3\), and what is the form of the solution to \(|x-1|<3\)?
\(x-1=3\) or \(x-1=-3\), so \(x=4\) or \(x=-2\). For \(|x-1|<3\): \(-3
(HL) How are even and odd functions defined, and what symmetry do they have?
Even: \(f(-x)=f(x)\), symmetric about the \(y\)-axis (e.g. \(x^2\), \(\cos x\)). Odd: \(f(-x)=-f(x)\), rotational symmetry about the origin (e.g. \(x^3\), \(\sin x\)).
(HL) What is a self-inverse function, and why restrict a domain to find an inverse?
Self-inverse: \(f=f^{-1}\), so the graph is symmetric in \(y=x\) (e.g. \(f(x)=\frac1x\)). A many-to-one function such as \(x^2\) needs a restricted domain (say \(x\ge0\)) to have an inverse (\(\sqrt x\)).

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