The Function Calculator is a versatile tool for evaluating, analyzing, and manipulating mathematical functions. It helps you work with various types of functions including polynomial, rational, exponential, logarithmic, trigonometric, and more complex composite functions.

Key Features

• Function Evaluation: Calculate f(x) for any given value of x, substituting the value into the function expression.
• Multiple Function Types: Support for polynomials, rational functions, exponentials, logarithms, trigonometric functions, and combinations.
• Function Operations: Perform operations like addition, subtraction, multiplication, and division of functions.
• Function Properties: Analyze zeros, intercepts, asymptotes, and critical points.
• Table Generation: Create tables of values for functions over specified intervals.
• Inverse Functions: Find and evaluate inverse functions when they exist.

Understanding Functions

A function is a mathematical relationship that assigns exactly one output value to each input value. The notation f(x) represents the output when x is the input.

Example 1: Polynomial Function

Evaluate f(x) = x³ - 4x² + 6x - 1 at x = 3:

f(3) = (3)<sup>3</sup> - 4(3)<sup>2</sup> + 6(3) - 1
     = 27 - 4(9) + 18 - 1
     = 27 - 36 + 18 - 1
     = 8

Example 2: Rational Function

Evaluate f(x) = (x² - 1)/(x + 2) at x = 3:

f(3) = (3<sup>2</sup> - 1)/(3 + 2)
     = (9 - 1)/5
     = 8/5
     = 1.6

Example 3: Composite Function

If f(x) = x² and g(x) = 2x + 1, evaluate (f ∘ g)(3):

(f ∘ g)(3) = f(g(3))

First find g(3):
g(3) = 2(3) + 1 = 7

Then find f(7):
f(7) = 7<sup>2</sup> = 49

Therefore: (f ∘ g)(3) = 49

Example 4: Piecewise Function

Evaluate the piecewise function:

       ⎧ x<sup>2</sup> + 1    if x < 0
f(x) = ⎨ 2x        if 0 ≤ x < 3
       ⎩ x - 1     if x ≥ 3

Evaluate at different points:
f(-2) = (-2)<sup>2</sup> + 1 = 5       (use first piece)
f(1) = 2(1) = 2             (use second piece)
f(5) = 5 - 1 = 4            (use third piece)

Example 5: Finding Function Zeros

Find the zeros of f(x) = x² - 5x + 6:

Set f(x) = 0:
x<sup>2</sup> - 5x + 6 = 0

Factor:
(x - 2)(x - 3) = 0

Zeros: x = 2 and x = 3

Verify:
f(2) = 4 - 10 + 6 = 0 ✓
f(3) = 9 - 15 + 6 = 0 ✓

• Order of Operations: Always follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) when evaluating.
• Substitution: Replace every occurrence of x with the given value, using parentheses: f(x) = x² - 3x becomes f(-2) = (-2)² - 3(-2).
• Domain Awareness: Check that your input value is in the function's domain before evaluating.
• Asymptotes: For rational functions, vertical asymptotes occur where the denominator equals zero.
• Function Notation: f(a + b) is not the same as f(a) + f(b) in general. Always substitute first, then evaluate.
• Zero vs. Undefined: f(x) = 0 means the output is zero; f(x) is undefined means there is no output for that input.
• Inverse Functions: To find f⁻¹(x), swap x and y, then solve for y. Not all functions have inverses.
• Even and Odd Functions: f(-x) = f(x) defines even functions (symmetric about y-axis); f(-x) = -f(x) defines odd functions (symmetric about origin).