Identity Function
Identity Function
The function f(x) = x. More generally, an identity function is one which does not change the domain values at all.
Note: This is called the identity function since it is the identity for composition of functions. That is, if f(x) = x and g is any function, then (f ° g)(x) = g(x) and (g ° f)(x) = g(x).
Key Formula
f(x)=x
Where:
- x = Any element in the domain; the output always equals this input
Worked Example
Problem: Let f(x) = x be the identity function and let g(x) = 3x + 2. Verify that composing f with g (in either order) returns g(x). Evaluate both compositions at x = 5.
Step 1: Compute (f ∘ g)(5). First apply g, then apply f to the result.
g(5)=3(5)+2=17,f(g(5))=f(17)=17
Step 2: Compute (g ∘ f)(5). First apply f, then apply g to the result.
f(5)=5,g(f(5))=g(5)=3(5)+2=17
Step 3: Compare both compositions to g(5) directly.
g(5)=17,(f∘g)(5)=17,(g∘f)(5)=17
Answer: Both (f ∘ g)(5) and (g ∘ f)(5) equal 17, which is the same as g(5). Composing any function with the identity function leaves it unchanged.
Another Example
Problem: The identity function has a distinctive graph. Describe it and find f(−4), f(0), and f(7).
Step 1: Apply the definition f(x) = x to each input value.
f(−4)=−4,f(0)=0,f(7)=7
Step 2: Plot the points (−4, −4), (0, 0), and (7, 7). Notice every point has the form (a, a).
Step 3: The graph is the straight line y = x, which passes through the origin with a slope of 1, making a 45° angle with the positive x-axis.
y=x,m=1,b=0
Answer: The identity function outputs −4, 0, and 7 respectively. Its graph is the line y = x through the origin with slope 1.
Frequently Asked Questions
Why is f(x) = x called the 'identity' function?
It is called the identity function because it acts as the identity element for function composition, just as 0 is the identity for addition and 1 is the identity for multiplication. Composing any function g with the identity function yields g itself: (f ∘ g)(x) = g(x) and (g ∘ f)(x) = g(x). The function 'identifies' each input with itself—nothing changes.
What is the domain and range of the identity function?
When defined on the real numbers, both the domain and range are all real numbers, (−∞, ∞). More generally, if the identity function is defined on any set S, its domain and range are both S, because every element maps to itself.
Identity function vs. Constant function
The identity function f(x) = x outputs a different value for each input—the output always equals the input. A constant function, such as f(x) = 5, outputs the same single value regardless of the input. The identity function's graph is a diagonal line with slope 1, while a constant function's graph is a horizontal line with slope 0. They are, in a sense, opposites: one preserves all information about the input, and the other discards it entirely.
Why It Matters
The identity function is fundamental in algebra and higher mathematics because it serves as the 'do nothing' operation for composition, much like adding zero or multiplying by one. When you study inverse functions, you check that f(f⁻¹(x)) = x, meaning the composition equals the identity function. It also appears in linear algebra as the identity matrix and in abstract algebra as the identity element of transformation groups.
Common Mistakes
Mistake: Confusing the identity function f(x) = x with a constant function like f(x) = 1.
Correction: The identity function returns whatever value is put in. f(3) = 3, not 1. The number 1 is the multiplicative identity, but the identity function is not the constant 1.
Mistake: Thinking the identity function changes the input in some way, such as taking the absolute value.
Correction: f(x) = |x| is not the identity function because f(−3) = 3 ≠ −3. The identity function must return the exact input, including its sign: f(−3) = −3.
Related Terms
- Function — General concept that the identity function is
- Composition — Operation for which identity function is neutral
- Identity — Broader concept across all operations
- Domain — Set of inputs, equals range for identity
- Inverse Function — Composing f and f⁻¹ yields the identity
- Linear Function — Identity function is linear with slope 1
- Constant Function — Contrasting function that ignores input