Variables and Substitution Revision Notes for HSC SSCE Mathematics Advanced

Variables and Substitution

Introduction

Functions are fundamental relationships in mathematics that connect input values to output values. Function notation provides a clear and efficient way to express these relationships, identify variables, and perform substitutions. This note explores how to use function notation, distinguish between independent and dependent variables, and substitute values into functions.

Function notation

Function notation is a mathematical shorthand for expressing relationships between variables. Instead of writing equations in the form $y = 2x + 3$ , we can use function notation to make the relationship more explicit.

In function notation, $f(x)$ represents the output value that corresponds to the input $x$ . The general form is:

$y = f(x)$

Here, $f(x)$ represents the output value for the input $x$ . The letter $f$ is the name of the function, and $x$ is the input variable.

infoNote

Reading Function Notation

The notation $f(x)$ is read as "f of x". It represents the unique output value that corresponds to a given input value $x$ . The expression is referred to as "the value of f at x".

For example, $f(4)$ represents the value of the function when $x = 4$ . If we have a function $f(x) = 2x + 3$ , then:

$f(x)$ is read as "f of x"
$f(4)$ means "the value of f when x equals 4"
To find $f(4)$ , substitute $x = 4$ into the function rule

lightbulbExample

Converting to Function Notation

Question: An equation is given by $y = 5x - 3$ . Rewrite the equation using function notation, where $f$ is the name of the function.

Solution:

Replace $y$ with the function notation $f(x)$ .

The function is $f(x) = 5x - 3$ .

Question: Use function notation to represent 'the value of $y$ at $x = 4$ '.

Solution:

To find the value of the function $f$ at a specific point, we replace the $x$ inside the brackets with the given value.

The value of $y$ at $x = 4$ is written as $f(4)$ .

Variables in function notation

Understanding the roles of different variables in functions is essential for working with function notation effectively. Variables have specific meanings depending on their position and function within the mathematical relationship.

Independent variable

The independent variable is the variable used to represent values in the domain (input values) of a function. This variable can be chosen freely, and its value does not depend on any other variable in the function.

In a function $f(x)$ , the letter $x$ is typically the independent variable. It represents the input that we can control or choose.

Dependent variable

The dependent variable is the variable used to represent the output values of a function. Its value depends on the input and the function's rule.

In the equation $y = f(x)$ , both $y$ and $f(x)$ represent the dependent variable. The value of $y$ or $f(x)$ is determined by:

The value of the independent variable (the input)
The rule that defines the function

infoNote

Variables in Context

A variable (algebraic) refers to quantities that are measurable or observable and are expected to either change over time or vary between individual observations.

Understanding the relationship

chatImportant

A function $f(x)$ describes a relationship in which each input produces exactly one output. The independent variable is chosen freely, while the dependent variable's value depends on the input and the function's rule.

Key relationships:

$x$ is the independent variable (input)
$y$ is the dependent variable (output)
The output value depends entirely on the input value and the function rule

For example, in the function $f(x) = 2x + 1$ :

$x$ is the independent variable
$f(x)$ (or $y$ ) is the dependent variable

lightbulbExample

Identifying Variables in Context

Question: The cost, $C$ , in dollars, to produce $n$ t-shirts is given by the function $C(n) = 12n + 50$ .

Part a: Identify the independent variable and determine what it represents in this context.

Solution:

The independent variable is the input variable of the function, which is the value that can be changed freely.

The independent variable is $n$ . It represents the number of t-shirts produced.

Part b: Identify the dependent variable and determine what it represents in this context.

Solution:

The dependent variable is the output variable of the function, whose value depends on the input.

The dependent variable is $C$ (or $C(n)$ ). It represents the total cost of production in dollars.

Part c: Explain why the cost is the dependent variable.

Solution:

The cost is the dependent variable because its value depends on the number of t-shirts that are produced. As the number of t-shirts changes, the total cost changes according to the function rule $C(n) = 12n + 50$ .

Substitution into functions

Substitution is the process of replacing a variable in an algebraic expression, formula, equation or function consistently with a particular value, another variable, expression or function.

Function notation makes substitution straightforward. To evaluate $f(a)$ for a function $f(x)$ , we replace $x$ with $a$ and simplify the resulting expression.

The general form is:

$f(a) = f(x) \text{ with } x \text{ replaced by } a$

Here, $a$ is the value or expression substituted for $x$ .

infoNote

Types of Substitution

Substitution works with both:

Numerical values: Replacing $x$ with a specific number
Algebraic expressions: Replacing $x$ with another variable or expression

lightbulbExample

Numerical Substitution

Question: For the function $f(x) = x^2 - 4x + 7$ , evaluate $f(3)$ .

Solution:

Substitute $x = 3$ into $f(x) = x^2 - 4x + 7$ and simplify the expression.

f(x) = x^2 - 4x + 7 \quad \text{(Write the function)} \\ f(3) = 3^2 - 4(3) + 7 \quad \text{(Substitute } x = 3\text{)} \\ f(3) = 9 - 12 + 7 \quad \text{(Evaluate the square and multiplication)} \\ f(3) = 4 \quad \text{(Evaluate)}

Check: Substituting a numerical value like 3 gives a numerical output.

lightbulbExample

Algebraic Substitution

Question: For the function $f(x) = x^2 - 4x + 7$ , evaluate $f(a + 2)$ .

Solution:

Substitute $x = a + 2$ into $f(x) = x^2 - 4x + 7$ and simplify the expression.

f(x) = x^2 - 4x + 7 \quad \text{(Write the function)} \\ f(a + 2) = (a + 2)^2 - 4(a + 2) + 7 \quad \text{(Substitute } x = a + 2\text{)} \\ f(a + 2) = a^2 + 4a + 4 - 4a - 8 + 7 \quad \text{(Expand the square and distribute)} \\ f(a + 2) = a^2 + 3 \quad \text{(Combine like terms)}

Check: Substituting an expression like $a + 2$ results in an algebraic expression, demonstrating the versatility of function notation.

bookmarkSummary

Key Points to Remember:

Function notation $f(x)$ represents the output for an input $x$ , and is used to evaluate functions at specific values
The independent variable (usually $x$ ) is the input that can be chosen freely
The dependent variable (usually $y$ or $f(x)$ ) is the output whose value depends on the input and the function's rule
Substitution means replacing the variable in a function with a specific value or expression, then simplifying
To evaluate $f(a)$ , substitute $a$ for $x$ in the function rule and simplify. This works for both numerical values and algebraic expressions

Variables and Substitution (HSC SSCE Mathematics Advanced): Revision Notes