Relearn Math as Adult A-Z(2): Functions
Summary
This blog post breaks down the concept of functions, making it accessible to adult learners looking to refresh or build their mathematical foundation. We cover fundamental definitions, explore various types of functions, and delve into transformations, expansion, and factorization, providing clear examples and practice problems along the way.
What is a Function?
At its core, a function is a rule that assigns a single, unique output to each input. Imagine a vending machine: you select an item (the input), and the machine dispenses only that item (the output). This one-to-one relationship is what defines a function.
Formal Definition
A function \(f\) from a set \(A\) (the domain) to a set \(B\) (the codomain), denoted as \(f: A \to B\), is a relation where each element \(x\) in \(A\) is associated with exactly one element \(y\) in \(B\). We write this as \(y = f(x)\), where \(f(x)\) represents the output for a given input \(x\). The range of \(f\) is the set of all actual output values: \(\{f(x) \mid x \in A\} \subseteq B\).
Key Concepts:
- Domain: The set of all possible input values (often denoted by \(x\)).
- Range: The set of all possible output values (often denoted by \(y\) or \(f(x)\)).
- Independent Variable: The input variable (usually \(x\)).
- Dependent Variable: The output variable (usually \(y\) or \(f(x)\)), whose value depends on the input.
Example
Consider the function \(f(x) = x + 5\).
- Input: \(x\)
- Output: \(f(x)\)
- If \(x = 2\), then \(f(2) = 2 + 5 = 7\).
Types of Functions
Functions come in many flavors! Here are some of the most common types you'll encounter:
- Linear Function: A function of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. It's a straight line when graphed.
- Quadratic Function: A function of the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\). Its graph is a parabola.
- Polynomial Function: A function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(n\) is a non-negative integer and the \(a_i\) are constants.
- Rational Function: A function that can be written as the ratio of two polynomials: \(f(x) = \frac{P(x)}{Q(x)}\), where \(Q(x) \neq 0\). Be mindful of potential domain restrictions!
- Exponential Function: A function of the form \(f(x) = a^x\), where \(a > 0\) and \(a \neq 1\).
- Logarithmic Function: The inverse of an exponential function, written as \(f(x) = \log_a x\), where \(a > 0\) and \(a \neq 1\). Remember that the domain is restricted to \(x > 0\).
- Trigonometric Functions: Functions like sine (\(sin(x)\)), cosine (\(cos(x)\)), and tangent (\(tan(x)\)) that relate angles of a triangle to the ratios of its sides.
Example
- Linear: \(f(x) = 4x - 1\)
- Quadratic: \(f(x) = x^2 + 3x - 2\)
- Exponential: \(f(x) = (1/2)^x\)
Transforming Functions
Transforming functions involves manipulating their graphs by shifting, stretching, compressing, or reflecting them.
Vertical Shifts
- Upward Shift: Adding a constant \(c\) to the function: \(f(x) + c\).
- Downward Shift: Subtracting a constant \(c\) from the function: \(f(x) - c\).
Example
Let \(f(x) = x^2\). Then \(f(x) + 3 = x^2 + 3\) shifts the parabola upwards by 3 units.
Horizontal Shifts
- Leftward Shift: Replacing \(x\) with \(x + c\) in the function: \(f(x + c)\).
- Rightward Shift: Replacing \(x\) with \(x - c\) in the function: \(f(x - c)\).
Example
Let \(f(x) = x^2\). Then \(f(x - 2) = (x - 2)^2\) shifts the parabola to the right by 2 units.
Vertical Stretching and Compression
- Vertical Stretch: Multiplying the function by a constant \(a > 1\): \(a \cdot f(x)\).
- Vertical Compression: Multiplying the function by a constant \(0 < a < 1\): \(a \cdot f(x)\).
Example
Let \(f(x) = x^2\). Then \(2f(x) = 2x^2\) stretches the parabola vertically, while \(0.5f(x) = 0.5x^2\) compresses it vertically.
Horizontal Stretching and Compression
- Horizontal Compression: Replacing \(x\) with \(ax\), where \(a > 1\): \(f(ax)\).
- Horizontal Stretch: Replacing \(x\) with \(ax\), where \(0 < a < 1\): \(f(ax)\).
Example
Let \(f(x) = \sqrt{x}\). Then \(f(2x) = \sqrt{2x}\) compresses the graph horizontally, while \(f(0.5x) = \sqrt{0.5x}\) stretches it horizontally.
Reflections
- Reflection about the x-axis: Multiplying the function by \(-1\): \(-f(x)\).
- Reflection about the y-axis: Replacing \(x\) with \(-x\): \(f(-x)\).
Example
Let \(f(x) = x^3\). Reflecting about the x-axis gives \(-f(x) = -x^3\). Reflecting about the y-axis gives \(f(-x) = (-x)^3 = -x^3\).
Expanding Functions
Expanding a function involves rewriting it by removing parentheses and simplifying the expression.
Example
Expand \(f(x) = (x - 1)(x + 2)\). \(f(x) = x^2 + 2x - x - 2 = x^2 + x - 2\)
More complex expansions can involve higher powers:
Example
Expand \(f(x) = (x + 1)^3\). \(f(x) = (x + 1)(x + 1)(x + 1) = (x^2 + 2x + 1)(x + 1) = x^3 + x^2 + 2x^2 + 2x + x + 1 = x^3 + 3x^2 + 3x + 1\)
Factorising Functions
Factorising a function is the opposite of expanding – it means expressing the function as a product of simpler factors.
Example
Factorise \(f(x) = x^2 - 5x + 6\). \(f(x) = (x - 2)(x - 3)\)
More complicated examples may require recognizing special patterns:
Example
- Factorise \(f(x) = 3x^2 + 9x\). Answer: \(f(x) = 3x(x + 3)\)
- Factorise \(f(x) = x^2 - 16\). Answer: \(f(x) = (x - 4)(x + 4)\) (Difference of Squares)
Factoring Techniques
- Greatest Common Factor (GCF): Always look for the largest factor common to all terms first.
- Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\).
- Perfect Square Trinomials: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\).
- Quadratic Formula: If you can't easily factor a quadratic, use the quadratic formula to find the roots and then construct the factors.
Practice Questions
Questions 1
Given \(f(x) = 5x - 3\), find \(f(2)\).
Answers
\(f(2) = 5(2) - 3 = 10 - 3 = 7\)
Questions 2
What is the domain of the function \(f(x) = \frac{1}{x - 5}\)?
Answers
The domain is all real numbers except \(x = 5\), since the denominator cannot be zero. In interval notation: \((-\infty, 5) \cup (5, \infty)\).
Questions 3
Transform the function \(f(x) = x^2\) by reflecting it across the x-axis and then shifting it 1 unit down. What is the new function?
Answers
Reflection across the x-axis: \(-x^2\). Shifting down 1 unit: \(-x^2 - 1\). So, the new function is \(g(x) = -x^2 - 1\).
Questions 4
Expand the function \(f(x) = (2x + 1)(x - 3)\).
Answers
\(f(x) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\)
Questions 5
Factorize the function \(f(x) = x^2 - 4x + 4\).
Answers
\(f(x) = (x - 2)(x - 2) = (x - 2)^2\)
Questions 6
Given the function \(f(x) = |x|\), what is the result after compressing it horizontally by a factor of 2?
Answers
Horizontal compression by a factor of 2 means replacing \(x\) with \(2x\). Therefore, the result is \(f(2x) = |2x|\).
Questions 7
If \(f(x) = x + 2\) and \(g(x) = x^2 - 1\), find \(f(g(x))\).
Answers
\(f(g(x)) = f(x^2 - 1) = (x^2 - 1) + 2 = x^2 + 1\)
Questions 8
The graph of \(f(x) = x\) is stretched vertically by a factor of 4 and then shifted 3 units to the left. Write the equation of the transformed function.
Answers
Vertical stretch by a factor of 4: \(4x\). Shifted 3 units to the left: \(4(x + 3)\). So the equation of the transformed function is \(g(x) = 4(x + 3) = 4x + 12\).