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Common Functions — Definition, Formula & Examples

Common functions are the standard families of functions — such as linear, quadratic, absolute value, square root, exponential, and logarithmic — that appear repeatedly throughout algebra, precalculus, and beyond. Recognizing each family by its equation and graph shape is a foundational skill for solving equations, modeling data, and understanding transformations.

A common function refers to any member of the set of elementary function families typically catalogued in precalculus: the identity function f(x)=xf(x)=x, constant function f(x)=cf(x)=c, quadratic f(x)=x2f(x)=x^2, cubic f(x)=x3f(x)=x^3, absolute value f(x)=∣x∣f(x)=|x|, square root f(x)=xf(x)=\sqrt{x}, reciprocal f(x)=1xf(x)=\tfrac{1}{x}, exponential f(x)=bxf(x)=b^x (with b>0, b≠1b>0,\,b\neq1), logarithmic f(x)=log⁡bxf(x)=\log_b x, and the trigonometric functions sin⁡x\sin x, cos⁡x\cos x, and tan⁡x\tan x. Each is defined by a distinct algebraic rule, domain, range, and characteristic graph shape.

How It Works

When you encounter a new equation or graph, your first move is to identify which common function family it belongs to. Once you know the parent function, you can predict its domain, range, intercepts, symmetry, and end behavior. You can then apply transformations — shifts, reflections, stretches — to match the specific equation you're working with. For example, g(x)=3∣x−2∣+1g(x) = 3|x - 2| + 1 is a transformed absolute value function: the parent is f(x)=∣x∣f(x)=|x|, shifted right 2, stretched vertically by 3, and shifted up 1. Memorizing the shapes and key features of each parent function saves significant time on graphing problems and function analysis.

Worked Example

Problem: Identify the parent function, domain, and range of h(x)=x−4+3h(x) = \sqrt{x - 4} + 3.
Step 1: Identify the parent function: The expression contains a square root, so the parent function is the square root function.
f(x)=xf(x) = \sqrt{x}
Step 2: Find the domain: The radicand must be non-negative, so set x−4≥0x - 4 \geq 0.
x≥4⇒Domain: [4,∞)x \geq 4 \quad \Rightarrow \quad \text{Domain: } [4, \infty)
Step 3: Find the range: The smallest value of x−4\sqrt{x-4} is 0 (when x=4x=4), so the smallest output is 0+3=30 + 3 = 3. Since the square root grows without bound, the range extends upward from 3.
Range: [3,∞)\text{Range: } [3, \infty)
Step 4: Describe the transformation: Compared to the parent f(x)=xf(x)=\sqrt{x}, the graph is shifted 4 units right and 3 units up.
Answer: Parent function: f(x)=xf(x)=\sqrt{x}. Domain: [4,∞)[4,\infty). Range: [3,∞)[3,\infty).

Another Example

Problem: Given g(x)=−2x2+8g(x) = -2x^2 + 8, identify the parent function and determine whether gg opens upward or downward, and state the vertex.
Step 1: Identify the parent function: The highest-degree term is x2x^2, so the parent function is the quadratic.
f(x)=x2f(x) = x^2
Step 2: Direction of opening: The leading coefficient is −2-2, which is negative, so the parabola opens downward.
Step 3: Find the vertex: Write g(x)=−2(x−0)2+8g(x) = -2(x-0)^2 + 8. The vertex is at (h,k)(h,k).
Vertex: (0, 8)\text{Vertex: } (0,\, 8)
Answer: Parent function: f(x)=x2f(x)=x^2. The parabola opens downward with vertex (0,8)(0, 8).

Why It Matters

In Algebra 2 and Precalculus, nearly every graphing and transformation problem starts by identifying the parent function. Physics and economics courses model real situations with exponential, quadratic, and linear functions, so recognizing these families on sight lets you choose the right model quickly. Standardized tests like the SAT and ACT regularly ask you to match equations to graph shapes, making fluency with common functions directly useful for college admissions.

Common Mistakes

Mistake: Confusing the square root function with the quadratic function when reading graphs.
Correction: A quadratic (x2x^2) is a full parabola symmetric about a vertical axis, while the square root (x\sqrt{x}) is only the top half of a sideways parabola starting at the origin and extending to the right. Check whether the curve has two branches (quadratic) or one (square root).
Mistake: Forgetting that the reciprocal function f(x)=1/xf(x)=1/x and the logarithmic function f(x)=log⁡xf(x)=\log x both have restricted domains.
Correction: The reciprocal function excludes x=0x=0 (domain: (−∞,0)∪(0,∞)(-\infty,0)\cup(0,\infty)), and the logarithmic function requires x>0x>0 (domain: (0,∞)(0,\infty)). Always check for values that make the function undefined.

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