Definition of a Function
A function is a rule that assigns exactly one output to each input. The set of allowed inputs is the domain; the set of possible outputs is the range. The one-output-per-input rule is strict: if an input produces two different outputs, it's not a function.
Vertical line test: on a graph, if any vertical line crosses the graph more than once, the relation is not a function. For example, a circle fails because a single x-value has two y-values.
Example: f(x) = 2x + 3 is a function. For each input x, there is exactly one output: f(5) = 2(5) + 3 = 13.
Domain and Range
The domain is the set of all valid inputs. Restrictions usually come from:
- Division by zero — exclude x-values where the denominator is zero.
- Even roots of negatives — exclude x-values that make the radicand negative.
- Logarithms of non-positives — the argument of a log must be positive.
The range is the set of outputs the function can produce. Often easiest to find from the graph or by analyzing end behavior.
Example: Find the domain of f(x) = \sqrt{x-2}. The radicand must be non-negative: x - 2 ≥ 0, so the domain is \{x \in \mathbb{R} : x \geq 2\}.
Function Notation
The notation f(x) — read "f of x" — names a function f and an input x. It does NOT mean multiplication. The parentheses group the input.
To evaluate a function, substitute the specific input for x everywhere. For example, if f(x) = 2x + 3, then f(5) replaces every x with 5: f(5) = 2(5) + 3 = 13.
Multiple functions often use different letters: f(x), g(x), h(x). Specific functions may use conventional names: sin(x), ln(x), e^x.
Types of Functions
Linear Functions
A linear function has the form y = mx + b, where m is the slope (rate of change) and b is the y-intercept. Its graph is always a straight line. Linear functions are the basis for solving systems of equations.
Quadratic Functions
Quadratic functions have the form f(x) = ax² + bx + c and graph as parabolas. The vertex is the turning point, and the parabola opens upward when a > 0, downward when a < 0. See the quadratic equations guide for solving and graphing techniques.
Rational Functions
Rational functions are ratios of polynomials and can have vertical asymptotes, horizontal asymptotes, and holes. See the rational functions guide for complete graphing procedures.
Exponential Functions
Exponential functions have the form f(x) = a·bˣ. They grow (or decay) at a rate proportional to their current value, modeling population, compound interest, and radioactive decay. See the exponents and logarithms guide.
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Try an interaction checkTransformations of Functions
A general transformation has the form g(x) = a \cdot f(b(x - h)) + k, where each parameter plays a specific role:
- h — horizontal shift: f(x-3) shifts the graph 3 units right
- k — vertical shift: f(x) + 3 shifts the graph 3 units up
- a — vertical stretch/compression; also controls reflection
- b — horizontal stretch/compression (inverse effect)
- Negative signs reflect: -f(x) reflects across the x-axis; f(-x) reflects across the y-axis
Key gotcha: horizontal transformations act opposite to the sign. f(x - 3) shifts RIGHT by 3, not left. f(2x) compresses horizontally by a factor of 2, not stretches. Vertical transformations act in the expected direction.
Inverse Functions
The inverse of a function f, written f^{-1}(x), reverses the rule: if f(3) = 7, then f⁻¹(7) = 3. Graphically, the inverse is the reflection of f across the line y = x.
Finding an inverse: swap x and y, then solve for y.
Example: Find the inverse of f(x) = 2x + 3:
One-to-one requirement: only functions that pass both the vertical AND horizontal line tests have inverses that are themselves functions. For others, you must restrict the domain (as with √x and x²).
Function Composition
Composition chains two functions so the output of one becomes the input of the other:
Read right-to-left: apply g first, then apply f to that result.
Example: If f(x) = x^2, \quad g(x) = x+1, then:
Composition is NOT commutative: (f∘g)(x) generally does not equal (g∘f)(x). However, a function and its inverse always compose to the identity: (f ∘ f⁻¹)(x) = x.
Common Mistakes
Thinking f(x) means f times x
f(x) is notation for "the function f evaluated at input x." It is not multiplication. This confusion causes errors throughout algebra.
Confusing horizontal and vertical transformations
Horizontal transformations act opposite to what you might expect: f(x-2) shifts right, not left. Vertical transformations are more intuitive.
Practice Problems
- If f(x) = 3x - 4, find f(2) and f^{-1}(x).
- Find the domain of f(x) = \dfrac{1}{x^2 - 9}.
- Find the domain of f(x) = \sqrt{5 - x}.
- If f(x) = x^2 and g(x) = x - 3, find (f \circ g)(x) and (g \circ f)(x).
- Describe the transformation from f(x) = x² to g(x) = -2(x-3)² + 5.
- Is the graph of a circle a function? Use the vertical line test.
Answers
- f(2) = 2; f⁻¹(x) = (x+4)/3
- x ≠ ±3 (all reals except 3 and -3)
- x ≤ 5
- (f∘g)(x) = (x-3)²; (g∘f)(x) = x² - 3 (different!)
- Reflect across x-axis, vertical stretch by 2, shift right 3, shift up 5
- No — a vertical line through the interior crosses twice
Related Guides
Frequently Asked Questions
What is a function in math?
A function is a rule that assigns exactly one output to each input. For every x-value in the domain, there is exactly one y-value. The vertical line test is a quick graphical check: if any vertical line crosses the graph more than once, it is not a function.
What is the difference between domain and range?
The domain is the set of all valid input values (x-values) for a function. The range is the set of all possible output values (y-values) the function produces. For example, f(x) = √x has domain [0, ∞) and range [0, ∞).
What does f(x) mean?
f(x) is function notation. It names the function (f) and shows the input variable (x). f(3) means "evaluate the function f at input 3." It does not mean f times x — this is one of the most common beginner misunderstandings.
What is a composite function?
A composite function applies one function to the result of another. Written as (f ∘ g)(x) = f(g(x)), it means "first apply g to x, then apply f to the result." The order matters: f(g(x)) is usually different from g(f(x)).
How do you find the inverse of a function?
To find the inverse, swap x and y in the equation and solve for y. The inverse function "undoes" what the original function does. Not all functions have inverses — a function must be one-to-one (pass the horizontal line test) to have an inverse.
What are transformations of functions?
Transformations shift, stretch, compress, or reflect a function's graph. Common transformations include vertical/horizontal shifts (adding constants), vertical/horizontal stretches (multiplying), and reflections (negating). Understanding transformations lets you graph any function family from its parent function.
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