Learning Objectives
  • Use function notation and evaluate functions accurately.
  • Identify domain and range in simple mappings and formulas.
  • Form and evaluate composite functions in the correct order.
  • Find inverse functions algebraically and verify them.
  • Interpret mapping diagrams and distinguish functions from non-functions.
Key Terms
Relation
A set of ordered pairs connecting elements of one set to another.
Function
A relation assigning each input exactly one output.
Domain
The allowed set of input values.
Range
The set of output values actually produced.
Composite function
A function formed by applying one function and then another.
Inverse function
A function that reverses the action of the original function.
One-to-one
A function in which different inputs produce different outputs.
What This Chapter Covers
  • Function notation and substitution
  • Domain and range
  • Mapping diagrams and ordered pairs
  • Composite functions
  • Inverse functions
Relations And Functions

A relation can pair inputs and outputs in any pattern. A function is more restrictive: every input in its domain must have exactly one output. Different inputs may share an output, but a single input cannot point to two different outputs.

Mapping pattern Function? Reason
Each input has one arrow Yes Every input has exactly one output
One input has two arrows No That input has two outputs
Two inputs share one output Yes Many-to-one is allowed
An input has no arrow No for the stated domain The function is not defined for every input
Function Notation

f(x) means the output of function f when the input is x. It does not mean f multiplied by x. To evaluate a function, substitute the input in brackets everywhere x appears.

Worked Example: Evaluate A Function

Question: Given f(x)=3x^2-2x+1, find f(-2).

  1. Substitute using brackets: f(-2)=3(-2)^2-2(-2)+1.
  2. Calculate the square before multiplying.
  3. 3(4)+4+1=17.

Answer: 17.

Domain And Range

The domain may be stated explicitly, shown in a mapping diagram or restricted by the formula. A denominator cannot be zero and a real square root cannot contain a negative number. The range consists of outputs produced from the allowed domain.

Worked Example: Finite Domain And Range

Question: Let f(x)=x^2-1 for domain \{-2,-1,0,1,2\}. Find the range.

  1. Calculate the outputs: 3, 0, -1, 0, 3.
  2. List distinct output values only.

Answer: Range =\{-1,0,3\}.

Composite Functions

gf(x)=g(f(x)) means apply f first and then g. The order matters: gf(x) is generally not equal to fg(x). Work from the function closest to x outward.

Worked Example: Composite Function

Question: Given f(x)=2x-3 and g(x)=x^2+1, find gf(x).

  1. First find f(x)=2x-3.
  2. Substitute this whole expression into g.
  3. g(f(x))=(2x-3)^2+1.
  4. Expand if required: 4x^2-12x+10.

Answer: gf(x)=4x^2-12x+10.

Worked Example: Numerical Composite

Question: Using the same functions, find fg(2).

  1. Apply g first: g(2)=5.
  2. Then apply f: f(5)=10-3=7.

Answer: 7.

Inverse Functions

An inverse reverses a one-to-one function. To find f^{-1}(x), write y=f(x), rearrange to make x the subject, and then replace y by x. The notation f^{-1} does not mean 1/f.

Finding An Inverse
  1. Write y=f(x).
  2. Rearrange the equation to make x the subject.
  3. Replace y by x in the final expression.
  4. Check by showing f(f^{-1}(x))=x or f^{-1}(f(x))=x.
Worked Example: Linear Inverse

Question: Find the inverse of f(x)=5x-7.

  1. Write y=5x-7.
  2. Add 7: y+7=5x.
  3. Divide by 5: x=(y+7)/5.
  4. Replace y with x.

Answer: f^{-1}(x)=\frac{x+7}{5}.

Worked Example: Fractional Linear Function

Question: Find the inverse of g(x)=\frac{x-4}{3}.

  1. Write y=(x-4)/3.
  2. Multiply by 3: 3y=x-4.
  3. Add 4: x=3y+4.

Answer: g^{-1}(x)=3x+4.

Inverse And Composite Relationship
f^{-1}(f(x))=x\qquad\text{and}\qquad f(f^{-1}(x))=x

These identities provide a powerful check. Domain restrictions may be needed, especially for quadratic functions, because a function must be one-to-one to have an inverse over the stated domain.

Solving Equations With Functions
Worked Example: Solve A Function Equation

Question: Given f(x)=4x+1, solve f(x)=29.

  1. Write 4x+1=29.
  2. Subtract 1 and divide by 4.

Answer: x=7.

Examination Guidance
  • Read gf(x) from right to left: f first, then g.
  • Use brackets when substituting an expression into another function.
  • List each range value once.
  • Check inverses by composition.
  • Distinguish f^{-1}(x) from the reciprocal 1/f(x).
Common Mistakes
  • Applying composite functions in the wrong order.
  • Dropping brackets when substituting a negative number or algebraic expression.
  • Calling a relation a function when one input has two outputs.
  • Including repeated values when listing a range.
  • Finding a reciprocal instead of an inverse function.
Knowledge Check And Practice

1. If f(x)=3x+2, find f(5).

Answer: 17.

2. If g(x)=x^2 and f(x)=x+1, find gf(3).

Answer: g(4)=16.

3. For the same functions, find fg(3).

Answer: f(9)=10.

4. Find the inverse of f(x)=2x-9.

Answer: f^{-1}(x)=\frac{x+9}{2}.

5. State the restriction on the domain of h(x)=1/(x-4).

Answer: x\ne4.

6. Why must a function be one-to-one to have an inverse function on its full domain?

Answer: Otherwise one output would have to reverse to more than one input, so the inverse would not be a function.
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