Certain functions appear so frequently in mathematics and its applications that they deserve special attention. This section surveys the most important elementary functions, including their graphs, domains, ranges, and key properties.
Quick Reference
| Function | Form | Domain | Range |
|---|---|---|---|
| Constant | f(x) = c | ℝ | {c} |
| Linear | f(x) = mx + b | ℝ | ℝ (if m ≠ 0) |
| Power (n even) | f(x) = xn | ℝ | [0, ∞) |
| Power (n odd) | f(x) = xn | ℝ | ℝ |
| f(x) = 1/x | Rational | ℝ – {0} | ℝ – {0} |
| f(x) = √x | Power (a = 1/2) | [0, ∞) | [0, ∞) |
| f(x) = ∛x | Power (a = 1/3) | ℝ | ℝ |
Constant Functions
A constant function assigns the same value c to every input: f(x) = c. Its graph is the horizontal line y = c.
Linear Functions
A linear function has the form f(x) = mx + b, where m (the slope) and b (the y-intercept) are fixed constants.
Key properties:
- m is the slope: positive m means the line rises left to right; negative m means it falls.
- If m = 0, the function reduces to the constant function f(x) = b.
- b is where the line crosses the y-axis: f(0) = b.
- If b = 0, the line passes through the origin.
- y and x are proportional (y = mx) if b = 0.
- y and x are inversely proportional (y = m/x) when their product is constant.
Power Functions
A power function has the form f(x) = xa, where a is a constant.
When a = n Is a Positive Integer
The graph of y = xn has two distinct shapes depending on whether n is even or odd.
When n is even
- The graph is U-shaped (like y = x2), symmetric about the y-axis (even function).
- Domain and range: ℝ and [0, ∞), respectively.
- Higher even powers are flatter near the origin and steeper for |x| > 1.
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When n is odd
- The graph is S-shaped, symmetric about the origin (odd function).
- Domain and range: both ℝ.
- Higher odd powers are flatter near the origin and steeper for |x| > 1.
When a = -1 or a = -2
The function f(x) = 1/x is odd, with domain ℝ – {0} and range ℝ – {0}. Its graph is symmetric about the origin and has two branches separated by the origin.
The function f(x) = 1/x2 is even, with domain ℝ – {0} and range (0, ∞).
Square Root and Cube Root
is defined only for x ≥ 0: domain [0, ∞), range [0, ∞). is defined for all x: domain ℝ, range ℝ. is an odd function; is neither odd nor even (its domain is not symmetric).
Polynomial Functions
A polynomial in x is a function of the form
where the exponents are nonnegative integers and an ≠ 0. We say f has degree n.
Special names by degree:
| Degree | Form | Name |
|---|---|---|
| 0 | y = a | constant |
| 1 | y = ax + b | linear |
| 2 | y = ax2 + bx + c | quadratic |
| 3 | y = ax3 + bx2 + cx + d | cubic |
| 4 | y = ax4 + bx3 + cx2 + dx + e | quartic |
End behavior is controlled by the leading term
- Even n, an > 0: both ends rise.
- Even n, an < 0: both ends fall.
- Odd n, an > 0: falls left, rises right.
- Odd n, an < 0: rises left, falls right.
Rational Functions
A rational function is a quotient of two polynomials:
Its domain excludes all values where Q(x) = 0.
Find the domain of
Solution
Find the roots of the denominator:
Algebraic and Transcendental Functions
Functions generated by finitely many additions, subtractions, multiplications, divisions, and rational-power operations are called algebraic functions. Polynomials, rational functions, and irrational functions (involving radical signs) are all algebraic.
A function that is not algebraic is called transcendental. The elementary transcendental functions include the trigonometric, inverse trigonometric, exponential, and logarithmic functions. We will explore these in subsequent chapters.
