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Year 12 Maths - Methods (Unit 3 & Unit 4) Power, exponential and logarithmic functions

Power functions

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Theory

A power function has the form \(y=x^{n}\), where the variable \(x\) is the base and the exponent \(n\) is a fixed number. The value of \(n\) — a positive integer, a negative integer, or a rational power — determines the domain, range, symmetry and overall shape of the graph. Even and odd powers give mirror-image and rotational symmetry, negative powers give asymptotes, and fractional powers give root curves.

A power function is any function of the form \(y=x^{n}\), where the variable \(x\) is the base and the exponent \(n\) is a fixed constant. A single number \(n\) fixes the domain, range, symmetry and shape, so power functions are grouped by the kind of number \(n\) is. Every power function passes through \((1,1)\).

For a positive integer \(n\), the graph also passes through the origin \((0,0)\). When \(n\) is even the curve is a parabola-type shape, symmetric about the \(y\)-axis, with range \([0,\infty)\); when \(n\) is odd it is a cubic-type curve with point symmetry about the origin and range \(\mathbb{R}\). Larger \(n\) makes the curve flatter near the origin and steeper beyond \(x=\pm 1\).

A negative integer power is a reciprocal, \(x^{-m}=\dfrac{1}{x^{m}}\): the \(y\)-axis \((x=0)\) and the \(x\)-axis \((y=0)\) are asymptotes, as for \(y=\dfrac{1}{x}\) and \(y=\dfrac{1}{x^{2}}\). A rational power is a root, \(x^{1/q}=\sqrt[q]{x}\); the square root \(x^{1/2}\) exists only for \(x\ge 0\), while the cube root \(x^{1/3}\) is defined for all real \(x\).

Key idea. Every \(y=x^{n}\) passes through \((1,1)\), and those with \(n>0\) also pass through \((0,0)\). For integer \(n\), the parity of \(n\) sets the symmetry: even \(n\) gives an even function (\(y\)-axis symmetry), odd \(n\) gives an odd function (origin symmetry).
Positive-integer power functions y=x, y=x^2 and y=x^3On one set of axes: y=x a straight line, y=x^2 an even parabola symmetric about the y-axis, and y=x^3 an odd cubic with point symmetry about the origin. All three pass through the origin and the point (1,1). x y y=x^2 y=x^3 y=x
\(y=x\), \(y=x^{2}\) (even) and \(y=x^{3}\) (odd): all pass through \((0,0)\) and \((1,1)\)
Negative-integer power functions y=1/x and y=1/x squaredy=1/x is an odd hyperbola with two opposite branches; y=1/x^2 is even with both branches above the x-axis. Both have the x-axis and y-axis as asymptotes. x y y=1/x y=1/x^2
\(y=\dfrac{1}{x}\) and \(y=\dfrac{1}{x^{2}}\): the axes are asymptotes; \(\dfrac{1}{x}\) is odd, \(\dfrac{1}{x^{2}}\) is even

The general form — a fixed exponent \(n\) applied to the variable \(x\):

\[y=x^{n}\qquad(0,0)\ \text{and}\ (1,1)\ \text{on the curve when }n>0\]
y=xn

Positive-integer powers — the range depends on the parity of \(n\):

\[n\ \text{even}:\ \text{range}=[0,\infty)\qquad n\ \text{odd}:\ \text{range}=\mathbb{R}\]
range=[0,∞)

Negative-integer powers — a reciprocal with the axes as asymptotes:

\[x^{-m}=\dfrac{1}{x^{m}}\quad(x\neq 0)\qquad \text{asymptotes: } x=0,\ y=0\]
x-m=1xm

Rational powers — a root; and the symmetry test for integer \(n\):

\[x^{1/q}=\sqrt[q]{x}\qquad x^{p/q}=\sqrt[q]{x^{\,p}}\]
x1/q=xq
\[f(-x)=f(x)\ (n\ \text{even})\qquad f(-x)=-f(x)\ (n\ \text{odd})\]
f(-x)=-f(x)
Anchor points. Every power curve passes through \((1,1)\); positive powers also pass through \((0,0)\). Use these, plus \((-1,1)\) for even \(n\) or \((-1,-1)\) for odd \(n\), to fix a quick sketch.

How to describe the graph of \(y=x^{n}\)

  1. Classify \(n\). Decide whether \(n\) is a positive integer (even or odd), a negative integer, or a rational power (a root).
  2. Domain. Positive powers: all real \(x\). Negative powers: \(x\neq 0\). Even roots such as \(x^{1/2}\): \(x\ge 0\); odd roots such as \(x^{1/3}\): all real \(x\).
  3. Symmetry and shape. For integer \(n\), even \(n\) gives an even function (\(y\)-axis symmetry), odd \(n\) gives an odd function (origin symmetry). Negative powers curve away from both axes.
  4. Key features. Mark \((0,0)\) (if \(n>0\)) and \((1,1)\); add any asymptotes \(x=0,\ y=0\) for a negative power.
  5. Range. Read it from the shape: e.g. \([0,\infty)\) for even positive \(n\), \(\mathbb{R}\) for odd positive \(n\), \((0,\infty)\) for \(1/x^{2}\).
Shortcut. To compare two powers, test a point in each interval: for \(0<x<1\) a higher power is smaller (e.g. \(x^{3}<x^{2}\)), while for \(x>1\) a higher power is larger (\(x^{3}>x^{2}\)).
Example 1 — even positive power
State the domain and range of \(y=x^{4}\), and say whether it is even or odd.
Solution
Domain — any real number can be raised to the fourth power:
domain\(=\)\(\mathbb{R}\)
Range — an even power is never negative:
\(x^{4}\)\(\ge\)\(0\)
range\(=\)\([0,\infty)\)
Symmetry — replace \(x\) with \(-x\):
\((-x)^{4}\)\(=\)\(x^{4}\)
\(\therefore\) domain \(\mathbb{R}\), range \([0,\infty)\); even (symmetric about the \(y\)-axis)
(-x)4=x4
Example 2 — negative power and asymptotes
For \(y=\dfrac{1}{x^{2}}=x^{-2}\), state the domain, range and asymptotes, and its symmetry.
Solution
Domain — the denominator cannot be zero:
\(x^{2}\)\(\neq\)\(0\ \Rightarrow\ x\neq 0\)
domain\(=\)\(\mathbb{R}\setminus\{0\}\)
Range — a square is positive, and \(\dfrac{1}{x^{2}}\) never reaches \(0\):
\(\dfrac{1}{x^{2}}\)\(>\)\(0\)
range\(=\)\((0,\infty)\)
Asymptotes and symmetry — test \(x\to -x\):
\((-x)^{-2}\)\(=\)\(x^{-2}\)
\(\therefore\) domain \(\mathbb{R}\setminus\{0\}\), range \((0,\infty)\); asymptotes \(x=0,\ y=0\); even
1x2>0
Example 3 — rational power (cube root)
The cube-root function is \(y=x^{1/3}\). State its domain and range, then find the value of \(x\) for which \(y=2\) and for which \(y=-2\).
Solution
Domain and range — every real number has a real cube root:
domain\(=\)\(\mathbb{R}\)
range\(=\)\(\mathbb{R}\ \ (\text{odd})\)
Solve \(x^{1/3}=2\) — cube both sides:
\(x\)\(=\)\(2^{3}=8\)
Solve \(x^{1/3}=-2\):
\(x\)\(=\)\((-2)^{3}=-8\)
\(\therefore\) domain \(\mathbb{R}\), range \(\mathbb{R}\); \(x=8\) when \(y=2\), and \(x=-8\) when \(y=-2\)
Rational-power functions y=square root of x and y=cube root of xy=x^(1/2) exists only for x>=0 and rises from the origin; y=x^(1/3) is defined for all real x and has point symmetry about the origin. Both pass through the origin and (1,1). x y y=x^(1/2) y=x^(1/3)
23=8
Example 4 — comparing two power functions
The curves \(y=x^{2}\) and \(y=x^{3}\) are both power functions. Find their points of intersection, and state the values of \(x\) for which \(x^{3}>x^{2}\).
Solution
Intersection — set the two equal and factorise:
\(x^{2}\)\(=\)\(x^{3}\)
\(x^{3}-x^{2}\)\(=\)\(0\)
\(x^{2}(x-1)\)\(=\)\(0\)
\(x\)\(=\)\(0\ \text{or}\ x=1\)
Compare — test the sign of \(x^{3}-x^{2}=x^{2}(x-1)\):
\(x^{2}\)\(\ge\)\(0\ \text{always}\)
\(x-1\)\(>\)\(0\ \Leftrightarrow\ x>1\)
\(\therefore\) they intersect at \((0,0)\) and \((1,1)\); \(x^{3}>x^{2}\) for \(x>1\)
x2(x-1)=0

Common pitfalls

The range depends on the parity of \(n\). \(y=x^{2}\) has range \([0,\infty)\), not all of \(\mathbb{R}\); only odd positive powers such as \(y=x^{3}\) have range \(\mathbb{R}\).
Even roots are restricted; odd roots are not. \(y=x^{1/2}\) is defined only for \(x\ge 0\), but \(y=x^{1/3}\) is defined for all real \(x\). Do not give the cube root the same restricted domain as the square root.
A negative power excludes \(x=0\). For \(y=\dfrac{1}{x}\) or \(y=\dfrac{1}{x^{2}}\) the value \(x=0\) is not in the domain and \(y=0\) is not in the range — the axes are asymptotes, never points on the curve.

Frequently asked questions

What is a power function?

A power function has the form \(y=x^{n}\), with the variable \(x\) as the base and \(n\) a fixed number — for example \(y=x^{2}\), \(y=\dfrac{1}{x}=x^{-1}\) and \(y=\sqrt{x}=x^{1/2}\). The value of \(n\) controls the whole shape of the graph.

How does the value of n change the shape of y = x to the n?

A positive even \(n\) gives a parabola-type curve on or above the \(x\)-axis; a positive odd \(n\) gives an increasing cubic-type curve through the origin. A negative \(n\) gives a reciprocal curve with the axes as asymptotes, and a rational power between \(0\) and \(1\) gives a root curve that rises steeply near the origin then flattens.

What are the domain and range of y = x squared and y = x cubed?

\(y=x^{2}\) has domain \(\mathbb{R}\) and range \([0,\infty)\), since squaring is never negative. \(y=x^{3}\) has domain \(\mathbb{R}\) and range \(\mathbb{R}\), since cubing a negative gives a negative. Both pass through \((0,0)\) and \((1,1)\).

Why is the square root function only defined for x greater than or equal to 0?

\(y=x^{1/2}\) asks for a number whose square is \(x\), and no real number squares to a negative, so it needs \(x\ge 0\). The cube root \(y=x^{1/3}\) is different: every real number has a real cube root, so it is defined for all \(x\).

What are the asymptotes of y = 1/x and y = 1/x squared?

Both have the \(y\)-axis \((x=0)\) as a vertical asymptote and the \(x\)-axis \((y=0)\) as a horizontal asymptote. \(y=\dfrac{1}{x}\) has branches in opposite quadrants, while \(y=\dfrac{1}{x^{2}}\) has both branches above the \(x\)-axis.

When is a power function even or odd?

For an integer power, \(y=x^{n}\) is even when \(n\) is even (graph symmetric about the \(y\)-axis) and odd when \(n\) is odd (graph with point symmetry about the origin). This is why even powers give mirror-image curves and odd powers give rotationally symmetric curves.

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