\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Functions

Definitions


A function is like a machine that produces an output from an input according to a rule.
To represent this machine, we write \(\textcolor{olive}{f}(\textcolor{colordef}{\text{input}}) = \textcolor{colorprop}{\text{output}}\). The brackets \((\) \()\) indicate the action of the function \(\textcolor{olive}{f}\) on the input.
We use function notation to name functions and their variables, replacing "\(\textcolor{colordef}{\text{input}}\)" by "\(\textcolor{colordef}{x}\)" and "\(\textcolor{colorprop}{\text{output}}\)" by "\(\textcolor{colorprop}{f(x)}\)".
We can write this function as
For example, if the rule is "twice the input":
we have \(\textcolor{olive}{f}(\textcolor{colordef}{x}) = \textcolor{colorprop}{2x}\):
When the input is \(\textcolor{colordef}{x} = \textcolor{colordef}{1}\), we get:$$\begin{aligned}\textcolor{olive}{f}(\textcolor{colordef}{1}) &= 2 \times \textcolor{colordef}{(1)}\\ &= \textcolor{colorprop}{2}\end{aligned}$$The table of values below shows the output values for different input values:

Definition Function
From an input value \(x\), a function \(f\) produces an output value \(f(x)\).
We can write:

  • \(f(x)\) is read as "\(f\) of \(x\)".
  • \(f(x)\) is called the image of \(x\).
Example
For \(f(x)=2x-1\) (the function that doubles the input and subtracts 1), find \(f(5)\).

\(\begin{aligned}[t] f(5)&=2\times (5)-1&(\text{substituting } x \text{ by } (5))\\&=9 \end{aligned}\)

Tables of Values

Definition Table of Values
The table of values for a function \(f\) provides a listing of pairs \((x, f(x))\), where \(x\) is an input value and \(f(x)\) is the corresponding output value produced by the function \(f\).
Example
For \(f(x)=x^2\), complete the following table:
\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
\(f(x)\)

  • \(\begin{aligned}[t] f(-2) &= (-2)^2 & (\text{substituting } x \text{ by } (-2)) \\ &= 4 \end{aligned}\)
  • \(\begin{aligned}[t] f(-1) &= (-1)^2 & (\text{substituting } x \text{ by } (-1)) \\ &= 1 \end{aligned}\)
  • \(\begin{aligned}[t] f(0) &= (0)^2 & (\text{substituting } x \text{ by } (0)) \\ &= 0 \end{aligned}\)
  • \(\begin{aligned}[t] f(1) &= (1)^2 & (\text{substituting } x \text{ by } (1)) \\ &= 1 \end{aligned}\)
  • \(\begin{aligned}[t] f(2) &= (2)^2 & (\text{substituting } x \text{ by } (2)) \\ &= 4 \end{aligned}\)
So the completed table is:
\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
\(f(x)\) \(4\) \(1\) \(0\) \(1\) \(4\)

Graphs

Definition Graph
A graph of a function is the set of all points \((\textcolor{colordef}{x},\textcolor{colordef}{f(x)})\) in the plane, where \(x\) is an input and \(f(x)\) is its output.
Method Plotting a Line Graph from a Table
To plot the graph of a function from a table of values, follow these steps:
  • Plot each point \((\textcolor{colordef}{x},\textcolor{colorprop}{f(x)})\) from the table onto the coordinate plane.
  • Connect the points with straight line segments.
Example
Here is a table of values for the function \(f(x) = x - 1\):
\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(f(x)\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
Plot the line graph of \(f\).

Plot the points \((-2, -3)\), \((-1, -2)\), \((0, -1)\), \((1, 0)\), \((2, 1)\), and \((3, 2)\). Then, connect the points with straight segments to form the line graph.

Reading Values and Solving \(f(x)=y\) on a Graph

Method Finding the value \(f(x)\) using a graph
To find \(f(2)\) on a graph, follow these steps:
  1. Locate the \(x\)-value: Find \(x = 2\) on the \(x\)-axis.
  2. Move vertically to the curve: From \(x = 2\), draw a vertical line up to the graph.
  3. Read the \(y\)-value: At the intersection with the curve, move horizontally to the \(y\)-axis to find the value \(f(2)\).
Thus, \(f(2) = 3\).
Method Finding \(x\) such that \(f(x) = y\) using a graph
To find \(x\) where \(f(x) = 3\) on this graph:
  1. Locate the \(y\)-value on the \(y\)-axis: Find \(3\) on the \(y\)-axis.
  2. Draw horizontally to the graph of the function: Draw a horizontal line from \(y = 3\) to the curve.
  3. Read the \(x\)-values: From the intersection points, draw vertical lines down to the \(x\)-axis and read the corresponding \(x\)-values.
Thus, the values of \(x\) for which \(f(x) = 3\) are \(x = 2\) and \(x = -2\).

Solving \(f(x)=y\) Algebraically

Method Solving \(f(x)=y\) algebraically
To find \(x\) such that \(f(x) = y\):
  • Write the equation \(f(x) = y\).
  • Solve for \(x\) using algebraic methods (e.g., inverse operations, isolating \(x\)).
Example
Let \(f(x) = 3x + 12\). Find all \(x\) such that \(f(x) = 0\).

We solve the equation:$$\begin{aligned} f(x) &= 0 \\ 3x + 12 &= 0 \\ 3x + 12 - 12 &= 0 - 12&& \color{gray}\text{(subtract 12 from both sides)} \\ 3x &= -12 \\ \frac{3x}{3} &= \frac{-12}{3} &&\color{gray}\text{(divide both sides by 3)} \\ x &= -4\\ \end{aligned}$$So the solution is \(x = -4\).
We can check this by calculating \(f(-4)\):$$f(-4) = 3 \times (-4) + 12 = -12 + 12 = 0$$So \(f(-4) = 0\), as expected.