Functions
Section: Algebra and graphs 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Function Notation, Domain and Range
Extended Only
A function is a rule that assigns exactly one output to each input. Function notation, such as f(x), names the rule and shows which input value it is applied to.
- f(x) means "the rule f applied to x" - to evaluate f(3), substitute x = 3 into the rule for f
- The domain of a function is the set of input values it is applied to
- The range is the resulting set of output values
- To find the range for a given domain, evaluate the function at every value in the domain
Worked Example: Evaluating a Function and Finding Its Range
- Question: f(x) = 3x + 4. The domain of f(x) is -3, 1, 5. Find the range of f(x).
- Step 1: f(-3)=3(-3)+4=-5
- Step 2: f(1)=3(1)+4=7
- Step 3: f(5)=3(5)+4=19
- Answer: range = -5, 7, 19
Common Mistakes
MistakeConfusing the domain (input values) with the range (output values), and giving the original x-values as the final answer
Fixthe range is always the set of results after applying the function - re-check which set the question is asking for
MistakeSubstituting a domain value into the wrong part of the function, e.g. adding before multiplying
Fixfollow the order of operations exactly as the function rule is written, multiplying by the coefficient of x before adding the constant
Inverse Functions
Extended Only
The inverse function, written f⁻¹(x), reverses the effect of f(x) - it takes an output of f back to its original input.
- To find f⁻¹(x) algebraically: write y = f(x), swap x and y, then rearrange to make y the subject again
- f⁻¹(a) is the input value that f maps to a - it can also be found directly by solving f(x) = a for x, without finding the general formula for f⁻¹(x) first
- Applying a function and then its inverse (in either order) returns the original value: f^-1(f(x))=x
Worked Example: Finding an Inverse Function
- Question: f(x) = 5x - 8. Find f⁻¹(x).
- Step 1: Let y=5x-8
- Step 2: Swap x and y: x=5y-8
- Step 3: Rearrange to make y the subject: x+8=5y y=(x+8/5)
- Answer: f^-1(x)=(x+8/5)
Worked Example: Finding an Inverse Function Value Directly
- Question: g(x)=2^(3/x), for x ≠ 0. Find g⁻¹(64).
- Step 1: g⁻¹(64) is the value of x for which g(x) = 64
- Step 2: Write 64 as a power of 2: 2^(3/x)=64=2^6
- Step 3: Equate the indices (since the bases are equal): (3/x)=6 x=(3/6)=0.5
- Answer: g⁻¹(64) = 0.5
Common Mistakes
MistakeForgetting to swap x and y before rearranging, and simply rearranging the original equation for x in terms of y
Fixthe swap step is essential - it is what makes the new equation the inverse, not just a rearrangement of the original
MistakeFinding the full formula for f⁻¹(x) first when only a single inverse value is needed, taking far longer than necessary
Fixto find f⁻¹(a) alone, it is often quicker to solve f(x) = a directly for x
Composite Functions
Extended Only
A composite function applies one function to the result of another. The order in which the functions are combined matters - fg(x) and gf(x) are generally different.
- fg(x) means f(g(x)) - apply g first, then apply f to the result
- gf(x) means g(f(x)) - apply f first, then apply g to the result
- To form a composite function algebraically, substitute the entire expression for the inner function into the outer function wherever x appears
Worked Example: Forming a Composite Function
- Question: f(x) = 2x - 1 and g(x) = x² + 3. Find fg(x) in the form ax² + b.
- Step 1: fg(x) means f(g(x)) - substitute g(x) into f: fg(x)=2(x^2+3)-1
- Step 2: Expand: 2x^2+6-1
- Answer: fg(x) = 2x² + 5
Worked Example: Evaluating a Composite Function
- Question: Using the same functions f(x) = 2x - 1 and g(x) = x² + 3, find gf(2).
- Step 1: Work from the inside out - find f(2) first: f(2)=2(2)-1=3
- Step 2: Substitute this result into g: g(3)=(3)^2+3=12
- Answer: gf(2) = 12
Common Mistakes
MistakeConfusing the order of fg(x) and gf(x), applying the outer function first instead of the inner one
Fixread composite notation from right to left - in fg(x), g is applied first (it is closest to x)
MistakeWhen evaluating a composite function numerically, substituting the number into both functions separately and combining the results incorrectly
Fixwork from the inside out - find the value of the inner function first, then substitute that result into the outer function
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