Functions and Rational Functions

A function is a rule that takes an input and gives back exactly one output. That definition decides which inputs are allowed, when a rule can be run backwards, and why some graphs have gaps.

What does f(x) mean?

f(x) is the output of the rule f when the input is x. It is not f multiplied by x. To find f(4), put 4 in place of every x in the rule.

If f(x) = 2x + 1, then f(3) = 2 × 3 + 1 = 7. The x inside the bracket is a placeholder, and a whole expression can take its place: f(a + 1) = 2a + 3.

See Function Notation.

How do you find the domain and range?

The domain is the set of inputs a function accepts. The range is the set of outputs it can produce.

To find a domain, start with every real number and remove the inputs that break the rule: a denominator cannot be zero, the inside of a square root cannot be negative, and the argument of a logarithm must be positive. $f\left(x\right)=\frac{1}{\left(x-2\right)}$f(x) = 1/(x − 2) has domain every real number except 2. $f\left(x\right)=\sqrt{x-5}$f(x) = √(x − 5) has domain $x\ge 5$x ≥ 5.

The range needs the graph. For $f\left(x\right)=x^2$f(x) = x² every input is allowed, but a square is never negative, so the range is $y\ge 0$y ≥ 0.

See Domain and Range.

Functions built out of pieces

A piecewise function uses a different rule on each stretch of its domain. To evaluate it, first find which stretch the input falls in, then use that rule. The boundary belongs to the rule whose inequality includes "or equal to".

The pieces join without a jump when both rules give the same value at the boundary. To make that happen, work out each rule at the boundary, set the two values equal, and solve for the unknown constant.

See Piecewise Functions, Graphing a Piecewise Function and Making a Piecewise Function Continuous.

Composite and inverse functions

A composite function sends the output of one function into another as its input. fg(x) means f(g(x)): apply g first, then f. If f(x) = x + 1 and $g\left(x\right)=x^2$g(x) = x², then fg(2) = f(4) = 5, but gf(2) = g(3) = 9.

An inverse function undoes the original: if f sends 3 to 10, then $f^{-1}$f⁻¹ sends 10 back to 3. To find one, reverse the order of the steps and replace each step with its opposite. f(x) = 3x + 1 multiplies by 3 and then adds 1, so $f^{-1}$f⁻¹ subtracts 1 and then divides by 3: $f^{-1}\left(x\right)=\frac{\left(x-1\right)}{3}$f⁻¹(x) = (x − 1)/3.

Swapping input and output swaps the coordinates of every point, so (3, 10) becomes (10, 3).

See Composite Functions and Inverse Functions.

How do transformations change a graph?

A change outside the bracket acts on the output and moves the graph the way it reads: $y=x^2+3$y = x² + 3 lifts every point up 3. A change inside the bracket acts on the input and moves the graph the opposite way: $y={\left(x-3\right)}^2$y = (x − 3)² shifts the graph 3 to the right, because x must be 3 bigger than before for the bracket to hold the same value.

Stretches follow the same split. $y=2x^2$y = 2x² doubles every height, and no point moves sideways. $y={\left(2x\right)}^2-4$y = (2x)² − 4 acts on the input, so the bracket reaches its old value at half the x, and the curve is squashed to half its width.

See Transforming Graphs, Stretching a Graph Vertically and Stretching a Graph Horizontally.

What makes exponential growth different?

Linear growth adds the same amount each step. Exponential growth multiplies by the same factor each step, so each step is bigger than the last, and the curve eventually overtakes any straight line. After n steps of doubling, a starting amount A has become $A\times 2^n$A × 2ⁿ.

Exponential decay is the same rule with a factor below 1. A half-life of 3 days means the amount halves every 3 days: 80 g becomes 40 g, then 20 g, then 10 g.

Nothing real doubles forever, because the room runs out. A logistic model grows like an exponential at first, then flattens onto a ceiling called the carrying capacity.

See Exponential Growth, Exponential Decay, Growth and Decay Problems and Logistic Growth and Carrying Capacity.

What is a logarithm for?

A logarithm asks what power the base was raised to: ${\log }_{2}8=3$log₂ 8 = 3 because $2^3=8$2³ = 8. Taking a logarithm undoes raising to a power.

The laws of logarithms come from the laws of exponents. Multiplying powers of one base adds the exponents, $2^3\times 2^4=2^7$2³ × 2⁴ = 2⁷, so log(ab) = log a + log b. Dividing subtracts: $\log \left(a/b\right)=\log a-\log b$log(a / b) = log a − log b. A power is repeated multiplying, so the exponent comes to the front: $\log \left(x^n\right)=n\log x$log(xⁿ) = n log x.

The power law solves an equation with the unknown in the exponent. Take the log of both sides, and the x comes down. To solve $3^x=20$3ˣ = 20: x log 3 = log 20, so $x=\log 20/\log 3\approx 2.727$x = log 20 / log 3 ≈ 2.727.

A calculator offers only ${\log }_{10}$log₁₀ and ln, and the change of base rule reaches every other base from those two: ${\log }_{2}N=\log N/\log 2$log₂ N = log N / log 2. ln is the logarithm with base $e\approx 2.718$e ≈ 2.718, and it obeys every law above.

See Exponentials and Logarithms, The Laws of Logarithms, Solving Exponential Equations, The Change of Base Rule and The Natural Logarithm.

Reading data on a log scale

A logarithmic axis places each value v at distance log v, so equal distances stand for equal multiples: 1, 10, 100 and 1000 are evenly spaced.

A log axis tests for exponential growth. If $y=a\times b^x$y = a × bˣ, taking logs of both sides gives log y = log a + x log b, a straight line in x. Plot log y against x: if the points straighten, the data is exponential, and undoing the logs on the gradient and the intercept recovers b and a.

The axis that takes the logs picks the model. A power law $y=a{x}^n$y = a xⁿ straightens when log y is plotted against log x, and the gradient is n. A logarithmic model y = a + b ln x straightens when y is plotted against ln x.

See Logarithmic Graphs, Straightening Growth with Logarithms, Straightening a Power Law and Fitting a Logarithmic Model.

Why do some graphs have gaps?

A rational function is one polynomial divided by another, and a fraction has no value where its denominator is zero. What the gap looks like depends on whether the factor that makes the denominator zero cancels.

A fraction is zero only when its numerator is zero, so the zeros of the numerator are the x-intercepts. Where the denominator is zero and nothing cancels, the graph has a vertical asymptote: just to one side the denominator is tiny, so the values grow without bound.

When the same factor appears above and below, it cancels, and the gap is a single missing point called a hole. $\frac{\left(x^2-4\right)}{\left(x-2\right)}=\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)}$(x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2). Cancel, and x + 2 is left for every x except 2, where the original reads 0 ÷ 0. A hole is a removable discontinuity; an asymptote cannot be repaired.

Far from the origin the leading terms decide the shape, so compare the degrees of the top and the bottom. If the bottom has the higher degree, the curve flattens onto y = 0. If the degrees are equal, it flattens onto the ratio of the leading coefficients: $\frac{\left(2x+1\right)}{\left(x-1\right)}$(2x + 1)/(x − 1) flattens onto y = 2. If the top is exactly one degree higher, the curve approaches a slanted line instead.

Long division finds that line. Divide polynomials the way you divide numbers: compare the leading terms, multiply back, subtract, and repeat. The quotient is the slant asymptote, and the remainder over the divisor fades to nothing far out. $\frac{\left(x^2+3x+5\right)}{\left(x+1\right)}=x+2+\frac{3}{\left(x+1\right)}$(x² + 3x + 5)/(x + 1) = x + 2 + 3/(x + 1), so the slant asymptote is y = x + 2.

See Zeros and Vertical Asymptotes, Holes Where a Factor Cancels, Horizontal Asymptotes by Degree, Long Division of Polynomials and Slant Asymptotes by Long Division.

The mistakes worth naming

• Reading f(x) as multiplication. f(4) means put 4 in for x, and f(a + b) is not f(a) + f(b) unless f(x) = kx.
• Shifting the wrong way. f(x + 3) moves the graph 3 to the left. A change inside the bracket works in reverse.
• Composing in the wrong order. fg(x) means f(g(x)): apply g first.
• Cancelling without recording the hole. Simplifying $\frac{\left(x^2-4\right)}{\left(x-2\right)}$(x² − 4)/(x − 2) to x + 2 loses the fact that x = 2 was never allowed.

Learn this properly in the app

Every lesson linked above is in Math Challenge, with an illustrated explanation and practice questions. Graphs of equations and coordinate geometry covers the lines and curves these transformations act on, and sequences and series continues from exponential growth into geometric series.