Functions & graphs

A function is a rule that assigns exactly one output to each allowed input. In high school, functions help you connect equations, tables, and graphs.

What is a function?

If \(f\) is a function, then \(f(x)\) means “the output when input is \(x\).” Example:

\[ f(x)=2x+3. \]

Then \(f(1)=5\), \(f(4)=11\), and \(f(-2)=-1\).

Domain and range

The domain is the set of allowed inputs. The range is the set of outputs produced from that domain.

Example 1. \(f(x)=\frac{1}{x-2}\)

• Domain: all real \(x\) except \(x=2\).
• Range: all real \(y\) except \(y=0\).

Example 2. \(g(x)=\sqrt{x+1}\)

• Domain: \(x \ge -1\) (square root needs non-negative inside).
• Range: \(y \ge 0\).

Function vs relation

Not every relation is a function. If one input gives two outputs, it is not a function.

On a graph, use the vertical line test: if any vertical line crosses the curve more than once, it is not a function.

Graph basics

To sketch a graph:

1. Choose key \(x\)-values and make a value table.
2. Plot points \((x,f(x))\) on labeled axes.
3. Connect appropriately (line, curve, or separate branches).
4. Identify intercepts and turning features.

Important function families

Linear: \(y=mx+c\)

Straight line with slope \(m\) and \(y\)-intercept \(c\). A larger \(|m|\) means steeper line.

Quadratic: \(y=ax^2+bx+c\)

Parabola. If \(a>0\), opens upward; if \(a<0\), opens downward.

Absolute value: \(y=|x|\)

V-shaped graph with vertex at the origin before transformations.

Rational: \(y=\frac{1}{x}\)

Hyperbola with asymptotes \(x=0\) and \(y=0\). Graph has two separate branches.

Transformations

Start from \(y=f(x)\):

• \(y=f(x)+k\): shift up by \(k\).
• \(y=f(x-h)\): shift right by \(h\).
• \(y=af(x)\): vertical stretch/compression by factor \(|a|\), and reflect if \(a<0\).
• \(y=f(bx)\): horizontal scale by factor \(\frac{1}{|b|}\), and reflect in \(y\)-axis if \(b<0\).

Example. From \(y=x^2\) to \(y=(x-3)^2+2\): move right 3, up 2.

Intercepts and meaning

• \(y\)-intercept: set \(x=0\).
• \(x\)-intercepts: solve \(f(x)=0\).

In applications, intercepts often represent starting value and break-even/zero-output points.

Piecewise functions

Some rules change in different intervals. Example:

\[ f(x)= \begin{cases} x+2, & x<1\\ 3, & x\ge 1 \end{cases} \]

Graph each rule only on its stated interval, and mark endpoints correctly (open/closed points).

Exam strategy

• State domain restrictions before graphing.
• Use a neat value table near turning points or asymptotes.
• Label key coordinates: intercepts, vertex, asymptote equations.
• Check whether a requested equation is function form or relation form.

History

The modern idea of a function developed over centuries. Early algebra connected formulas to unknown quantities, while analytic geometry (Descartes) linked equations to curves on coordinate axes.

Later, Euler and others formalized function notation such as \(f(x)\). Today, function thinking is central in mathematics, science, economics, and computing because it models how one quantity depends on another.

Derivation and reasoning

From two points to line equation

If a line passes through \((x_1,y_1)\) and \((x_2,y_2)\), slope is: \[ m=\frac{y_2-y_1}{x_2-x_1}. \] Then use \(y=mx+c\) and substitute one point to find \(c\).

Why vertex form helps for quadratics

Writing \(y=a(x-h)^2+k\) shows the turning point immediately: vertex \((h,k)\). This is why completing the square is useful for sketching and transformations.

Checkpoints

For \(f(x)=3x-2\), find \(f(5)\) and \(f(-1)\).

Answer: \(f(5)=13\), \(f(-1)=-5\).

State domain and range of \(g(x)=\sqrt{x-4}\).

Answer: Domain \(x\ge 4\), range \(y\ge 0\).

Is \(x=y^2\) a function of \(x\)? Use the vertical line test idea.

Answer: No.

For a positive \(x\), there are two \(y\)-values (\(+\sqrt{x}\) and \(-\sqrt{x}\)), so one \(x\) maps to two outputs.

For \(y=(x+1)^2-3\), state the transformation from \(y=x^2\).

Answer: Shift left 1 and down 3.

Find intercepts of \(y=x^2-4x+3\).

Answer: \(y\)-intercept: \((0,3)\).

Answer: \(x\)-intercepts: solve \(x^2-4x+3=0\Rightarrow(x-1)(x-3)=0\), so \((1,0)\), \((3,0)\).

Sketch idea: \(f(x)=\frac{1}{x-2}+1\). What are its asymptotes?

Answer: Vertical asymptote \(x=2\), horizontal asymptote \(y=1\).

Applications

Finance and budgeting

Linear functions model plans with fixed charges plus per-unit costs, such as phone plans or transport fares.

Motion and science

Position-time or velocity-time graphs are function graphs. Intercepts, gradients, and curvature describe physical behavior.

Data modeling

Exponential, quadratic, and rational models are used to approximate trends, optimize designs, and make predictions from measured data.

References

• Standard high-school algebra curricula on functions, graphs, and transformations.
• Analytic geometry and precalculus references covering domain/range and graph families.