2.2 Functions - AHL content

Learning objectives

—AHL 2.7 (HL)—Composite and inverse functions• Use composite functions in context: (f o g)(x)=f(g(x)).• Find inverse functions, including necessary domain restrictions.—AHL 2.8 (HL)—Graph transformations• Apply translations, reflections, vertical/horizontal stretches and composite transformations.• Perform transformations on SL/AHL model functions and real-context functions.• Understand order of transformations.—AHL 2.9 (HL)—Further modelling functions• Model with half-life exponential, natural logarithmic, sinusoidal, logistic and piecewise functions.• For sinusoidal models, use radians unless degrees are specified; phase shift is horizontal translation.• Logistic models use carrying capacity L; piecewise models may require continuity of parameters.—AHL 2.10 (HL)—Logarithmic scaling and linearization• Scale very large/small numbers using logarithms.• Linearize data with logarithms to test exponential or power relationships.• Interpret log-log and semi-log graphs; drawing/sketching them is not required in exams.

The composite (f∘g)(x)=f(g(x)) applies g first and then f. Its domain is restricted wherever the inner output is outside the outer function's domain.

Keep the order visible. If f(x)=√x and g(x)=x−1, then (f∘g)(x)=√(x−1), so x≥1; reversing the order gives √x−1 with a different domain and meaning.

For f(x)=2x+1 and g(x)=x², (f∘g)(3)=2·9+1=19, while (g∘f)(3)=7²=49. The functions need not commute.

Composition is not multiplication. Check which function acts first and state the domain rather than assuming both orders are valid.

Inverse workflow: restrict the original domain until the function is one-to-one, write $y=f(x)$ , swap $x$ and $y$ , and solve for $y$ . Then $(f\circ f^{-1})(x)=x$ on the range of $f$ and $(f^{-1}\circ f)(x)=x$ on the restricted domain. For $f(x)=(x-3)^2$ restricted to $x\ge3$ , $f^{-1}(x)=3+\sqrt{x}$ ; choosing $x\le3$ would give the other branch $3-\sqrt{x}$ .

For y=f(x), adding outside the function changes outputs: y=f(x)+b shifts vertically. Changing the input changes where an existing feature occurs: y=f(x−a) shifts right by a.

A factor outside, y=af(x), stretches or reflects vertically; a factor inside, y=f(bx), changes horizontal scale by 1/|b| and may reflect when b<0. Composite transformations must be applied in the stated order.

From y=x², the graph y=2(x−3)²+1 has vertex (3,1), opens upward and is twice as steep vertically. Read the vertex after the horizontal shift, then the vertical changes.

Inside and outside signs behave differently: f(x−3) moves right, while f(x)+3 moves up. Do not reverse a horizontal scale factor.

AHL models extend the SL families: half-life uses exponential decay; $a+b\ln x$ models logarithmic change; $a\sin(b(x-c))+d$ models cycles; $L/[1+Ce^{-kx}]$ models growth limited by carrying capacity $L$ ; piecewise functions use different rules on stated intervals.

For a half-life $h$ , $Q(t)=Q_0(1/2)^{t/h}$ . In a sinusoidal model, radians are assumed unless degrees are marked, the period is $2\pi/|b|$ , and $c$ is a horizontal translation or phase shift. In a logistic model, $L$ is the horizontal asymptote and carrying capacity.

Example

If a quantity starts at 800 and has half-life 3 years, $Q(t)=800(1/2)^{t/3}$ , so $Q(6)=200$ . For a piecewise model, solve shared-boundary parameter conditions when continuity is required; the formal definition of continuity is not required.

A model family is chosen from mechanism and shape, not fit alone. Do not use degrees in a radian model, call $L$ the initial value of a logistic curve, or assume two piecewise rules join continuously without checking their boundary values.

A logarithmic axis spaces equal ratios equally. Taking logs can turn a multiplicative model into a linear one: y=Ax^k gives log y=log A+k log x, while y=Ab^x gives log y=log A+x log b.

The slope on a log–log plot estimates a power k; the slope on a semi-log plot estimates log b. Choose the transformation that matches the proposed mechanism and keep the same log base when interpreting intercepts.

If doubling x roughly quadruples y, a log–log slope near 2 is more informative than a straight-line fit on ordinary axes. The intercept gives the scale factor only after undoing the logarithm.

A straight line after transformation does not prove the original relationship. Check units, zero/negative values and back-transform predictions before making the contextual claim.

Logarithms compress very large or small positive values into a manageable scale. On a semi-log plot, linearity supports $y=Ab^x$ and slope is $\log b$ ; on a log–log plot, linearity supports $y=Ax^k$ and slope is $k$ , with intercept $\log A$ . Students interpret these graphs in examinations but are not required to draw or sketch them. Zero and negative values cannot be logged without redefining the model.

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—IB Mathematics: Applications and InterpretationRead

2.2 Functions - AHL content

Learning objectives

Composition means apply one function inside another

Graph transformations act on input or output

Further models match decay, cycles, saturation and changing rules

Example

Log scales reveal multiplicative structure

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