Strand 5 — Functions & Calculus
Functions
5th Year · 6th Year (Leaving Cert)
- ✓By the end of this lesson students will be able to define and identify the domain and range of various functions.
- ✓By the end of this lesson students will be able to determine if a function is injective, surjective, or bijective, and find its inverse if it exists.
- ✓By the end of this lesson students will be able to form and evaluate composite functions.
- ✓By the end of this lesson students will be able to recognise and work with polynomial, exponential, logarithmic, and trigonometric functions.
- ✓By the end of this lesson students will be able to apply the laws of indices and logarithms to solve equations involving exponential and logarithmic functions.
Key concepts
A function is a rule that assigns to each input value (from its domain) exactly one output value. It can be thought of as a mapping from one set (the domain) to another set (the codomain).
The domain of a function is the complete set of all possible input values (x-values) for which the function is defined. For real functions, this often means avoiding division by zero or taking the square root of a negative number.
The range of a function is the complete set of all possible output values (y-values or f(x) values) that the function can produce for the given domain.
A function f is injective if every distinct element in its domain maps to a distinct element in its range. In other words, if f(x₁) = f(x₂), then x₁ = x₂. Graphically, it passes the horizontal line test.
A function f is surjective if every element in its codomain is the image of at least one element in its domain. This means the range of the function is equal to its codomain.
A function is bijective if it is both injective (one-to-one) and surjective (onto). A bijective function has a unique inverse function.
If a function f is bijective, then an inverse function, denoted f⁻¹(x), exists. The inverse function 'reverses' the action of f, meaning if f(a) = b, then f⁻¹(b) = a. The domain of f is the range of f⁻¹, and the range of f is the domain of f⁻¹.
A composite function is formed when the output of one function becomes the input of another function. The composition of f with g is denoted (f ∘ g)(x) and is defined as f(g(x)). The composition of g with f is (g ∘ f)(x) = g(f(x)).
A polynomial function is a function that can be expressed in the form of a polynomial. The domain of all polynomial functions is the set of all real numbers, ℝ.
An exponential function is a function where the variable appears as an exponent. The base 'a' must be a positive real number and not equal to 1. The natural exponential function uses Euler's number 'e' as its base.
A logarithmic function is the inverse of an exponential function. log_a(x) asks 'to what power must 'a' be raised to get 'x'?'. The domain of log_a(x) is x > 0. The natural logarithm uses base 'e' and is denoted ln(x).
These functions relate an angle of a right-angled triangle to the ratios of two side lengths. Common trigonometric functions include sine, cosine, and tangent.
Rules for simplifying expressions involving powers.
Rules for simplifying expressions involving logarithms.
Key facts to remember
- 1A function maps each input to exactly one output.
- 2The domain is the set of all valid inputs; the range is the set of all possible outputs.
- 3An inverse function f⁻¹(x) exists if and only if f(x) is bijective (one-to-one and onto).
- 4The domain of f is the range of f⁻¹, and the range of f is the domain of f⁻¹.
- 5The composition (f ∘ g)(x) means f(g(x)), applying g first, then f.
- 6For log_a(x) to be defined, x must be greater than 0.
- 7The laws of indices and logarithms are essential for solving exponential and logarithmic equations.
- 8e is Euler's number, an important mathematical constant approximately equal to 2.718.
Worked examples
Example 1
Let f(x) = (3x - 2) / (x + 1). Find the domain and range of f(x). Find f⁻¹(x) and state its domain and range.
Answer
Domain of f(x): ℝ \ {-1} Range of f(x): ℝ \ {3} f⁻¹(x) = (x + 2) / (3 - x) Domain of f⁻¹(x): ℝ \ {3} Range of f⁻¹(x): ℝ \ {-1}
Remember that the domain of a function is the range of its inverse, and vice versa. Always check for values that would make the denominator zero or lead to the square root of a negative number.
Example 2
Given the functions f(x) = 2x + 3 and g(x) = x² - 1. Find (f ∘ g)(x) and (g ∘ f)(x). Evaluate (f ∘ g)(2).
Answer
(f ∘ g)(x) = 2x² + 1 (g ∘ f)(x) = 4x² + 12x + 8 (f ∘ g)(2) = 9
Note that (f ∘ g)(x) is generally not equal to (g ∘ f)(x). The order of composition matters.
Example 3
Solve for x: log₂(x + 3) - log₂(x - 1) = 2.
Answer
x = 7/3
Always check your solution against the domain restrictions of the original logarithmic equation to ensure it is a valid solution.
Common mistakes
- ✗Confusing the domain and range, or incorrectly identifying restrictions for them (e.g., forgetting x > 0 for log functions).
- ✗Incorrectly applying the laws of logarithms or indices, such as assuming log(x + y) = log(x) + log(y).
- ✗Assuming every function has an inverse, or not checking if the function is bijective before attempting to find an inverse.
- ✗Mixing up the order of functions in a composite function, i.e., calculating g(f(x)) instead of f(g(x)) when asked for (f ∘ g)(x).
- ✗Failing to check if a solution to a logarithmic equation falls within the function's domain, potentially leading to extraneous solutions.
Exam tips
- ★Always state the domain and range of functions clearly, especially when asked, and be mindful of restrictions (e.g., denominators ≠ 0, arguments of logs > 0).
- ★When finding an inverse function, verify your answer by checking if f(f⁻¹(x)) = x or f⁻¹(f(x)) = x.
- ★Practice the laws of indices and logarithms regularly. They are fundamental for solving equations in this topic.
- ★For composite functions, work from the inside out: evaluate the inner function first, then apply the outer function to that result.
- ★Show all steps in solving equations, especially for logarithmic and exponential equations, as partial credit is awarded for correct method.
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