2.2 Functions - AHL content

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.