Algebra 1.2

Algebra · 1.2  |  Functions — Complete Knowledge Points

Chapter 1 · Algebra

1.2 Functions —
Complete Knowledge Points

§ 6 Topics · Domain · Composition · Inverse · Even & Odd · Common Functions

Contents

1. What is a Function?
2. Domain and Range
3. Function Composition
4. Inverse Functions
5. Even and Odd Functions
6. Common Functions

01

What is a Function?

A function is a rule that assigns exactly one output to every input. Think of it like a machine:

        ┌──────────┐
  x ──▶ │  f(x)    │ ──▶  output
        └──────────┘
        

The input is called the argument, the output is called the value, and we write it as $f(x)$, read as "f of x".

⚠️

Key rule: One input must never produce two different outputs. However, two different inputs can produce the same output.

02

Domain and Range

Term	Meaning
Domain	All valid input values ($x$)
Range	All possible output values ($f(x)$)

How to Find the Domain

Two main restrictions to watch for:

Rule 1 — No zero in the denominator

$$f(x) = \frac{1}{x - 3} \implies x \neq 3 \implies \text{Domain: } (-\infty, 3) \cup (3, +\infty)$$

Rule 2 — No negative number under a square root

$$f(x) = \sqrt{x - 2} \implies x - 2 \geq 0 \implies x \geq 2 \implies \text{Domain: } [2, +\infty)$$

Both rules combined

$$f(x) = \frac{1}{\sqrt{x-2}} \implies x - 2 > 0 \implies x > 2 \implies \text{Domain: } (2, +\infty)$$

💡

Interval notation reminder:
$[a, b]$ means $a \leq x \leq b$ — endpoints included
$(a, b)$ means $a < x < b$ — endpoints excluded

03

Function Composition

Function composition means plugging one function into another. You evaluate from the inside out — compute $g(x)$ first, then feed the result into $f$.

$$(f \circ g)(x) = f(g(x))$$

Example

Let $f(x) = x^2$ and $g(x) = x + 1$:

f ∘ g

$$f(g(x)) = f(x+1) = (x+1)^2$$

g ∘ f

$$g(f(x)) = g(x^2) = x^2 + 1$$

⚠️

Order matters: $f(g(x)) \neq g(f(x))$ in general.

Domain of a Composition

The domain of $f(g(x))$ requires two conditions to hold simultaneously:

1. $x$ must be in the domain of $g$
2. $g(x)$ must be in the domain of $f$

Example: Let $f(x) = \sqrt{x}$ and $g(x) = x - 3$, so $f(g(x)) = \sqrt{x-3}$.

$g(x)$ is defined for all $x$ ✅  ·  $f(g(x))$ requires $x - 3 \geq 0 \implies x \geq 3$

$$\text{Domain of } f(g(x)): [3, +\infty)$$

04

Inverse Functions

The inverse function $f^{-1}(x)$ reverses what $f(x)$ does:

$$f(a) = b \iff f^{-1}(b) = a$$

f a ─────────▶ b a ◀───────── b f⁻¹

How to Find an Inverse Function

1. Write $y = f(x)$
2. Swap $x$ and $y$
3. Solve for $y$
4. Write $y$ as $f^{-1}(x)$

Example: Find the inverse of $f(x) = 2x + 3$

Step-by-step

$$y = 2x + 3$$ $$x = 2y + 3 \quad \text{(swap } x \text{ and } y\text{)}$$ $$y = \frac{x - 3}{2}$$ $$f^{-1}(x) = \frac{x-3}{2}$$

Key Properties of Inverse Functions

Property	Description
Domain / Range swap	Domain of $f$ = Range of $f^{-1}$, and vice versa
Cancellation	$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
Graph reflection	Graph of $f^{-1}$ is the reflection of $f$ across $y = x$

⚠️

Not every function has an inverse. A function only has an inverse if it is one-to-one — every output comes from exactly one input. You can verify this with the Horizontal Line Test: if any horizontal line crosses the graph more than once, the function is not one-to-one.

05

Even and Odd Functions

These describe the symmetry of a function's graph.

Even Function

$$f(-x) = f(x) \quad \text{for all } x$$

The graph is symmetric about the y-axis. Example: $f(x) = x^2$, because $(-x)^2 = x^2$ ✅

Odd Function

$$f(-x) = -f(x) \quad \text{for all } x$$

The graph is symmetric about the origin. Example: $f(x) = x^3$, because $(-x)^3 = -x^3$ ✅

💡

Most functions are neither even nor odd. For example, $f(x) = x^2 + x$ is neither.

	Even	Odd
Condition	$f(-x) = f(x)$	$f(-x) = -f(x)$
Symmetry	y-axis	Origin
Example	$x^2$, $\cos x$	$x^3$, $\sin x$

06

Common Functions You Must Know

6.1  Linear Function

$$f(x) = mx + b$$

Graph: a straight line.  $m$ = slope, $b$ = y-intercept.

6.2  Quadratic Function

$$f(x) = ax^2 + bx + c \quad (a \neq 0)$$

Graph: a parabola. Opens up if $a > 0$, opens down if $a < 0$. Vertex at $x = -\dfrac{b}{2a}$.

6.3  Polynomial Function

$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$$

Defined for all real numbers. Degree $n$ determines the general shape.

6.4  Rational Function

$$f(x) = \frac{P(x)}{Q(x)}$$

Undefined where $Q(x) = 0$. May have vertical asymptotes where $Q(x) = 0$ and horizontal asymptotes depending on degrees of $P$ and $Q$.

6.5  Absolute Value Function

$$f(x) = |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$$

Graph: V-shape, vertex at the origin. Always outputs a non-negative value.

✓

Summary Checklist

• ✓ Understand what makes something a valid function
• ✓ Find the domain and range of a given function
• ✓ Compute $f(g(x))$ and $g(f(x))$ and understand they are different
• ✓ Find the domain of a composed function
• ✓ Find the inverse of a function algebraically
• ✓ Verify an inverse using $f(f^{-1}(x)) = x$
• ✓ Determine whether a function is even, odd, or neither
• ✓ Recognize and describe all 5 common function types