Functions are the building blocks of calculus. They describe relationships between variables, allowing us to model real-world phenomena. Understanding , types, and evaluation is crucial for solving complex problems in calculus.
and define where functions operate. Graphing functions reveals their behavior, including key features like intercepts and . Identifying helps solve equations and understand behavior, setting the stage for more advanced calculus concepts.
Function Fundamentals
Function notation evaluation
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Function notation [f(x)](https://www.fiveableKeyTerm:f(x)) represents the output value when the is x
Evaluate f(a) by substituting a for x in the function's equation and simplify
Function types and their notation
f(x)=mx+b with slope m and b
f(x)=ax2+bx+c with constants a, b, and c
f(x)=anxn+an−1xn−1+…+a1x+a0 with constants ai
f(x)=Q(x)P(x) with polynomial functions P(x) and Q(x)
f(x)=ax with positive constant a
f(x)=loga(x) with positive constant a=1
sin(x), cos(x), tan(x), etc.
defined by different equations over different intervals of the
Domain and range identification
Domain: Set of all possible input values (usually x) for a function
Most functions have a domain of all real numbers, unless restricted
Rational functions exclude x values that make the denominator zero
Square root functions require a non-negative argument under the square root
Logarithmic functions require a positive argument
: Set of all possible output values (usually y) for a function
Determined by the function's equation and domain restrictions
Linear functions have a range of all real numbers
Quadratic functions have a range depending on the parabola direction (up or down)
Exponential functions always have a positive range
Logarithmic functions have a range of all real numbers
Trigonometric functions have a limited range (e.g., −1≤sin(x)≤1)
Function graphing and features
Key features of function graphs
(zeros) where the graph crosses the x-axis (y=0)
y-intercept where the graph crosses the y-axis (x=0)
across the y-axis (), origin (), or other lines
Asymptotes that the graph approaches as x or y approaches infinity or a specific value
occur at x values where the function is undefined (e.g., denominators equal to zero)
occur when the function approaches a constant value as x approaches positive or negative infinity
where the function is increasing or decreasing
where the function reaches a highest or lowest value within a specific interval
indicating the direction of the curve (upward or downward)
where the concavity changes
Zeros of functions
Zeros () of a function are the x values where f(x)=0
Algebraic methods
: Factor the function and set each factor equal to zero
: For quadratic functions, use x=2a−b±b2−4ac
: For polynomial functions, list potential rational zeros and test using substitution
Graphical methods
Identify the x-intercepts of the function's graph
Function Representations and Operations
Function representations
Tables list input (x) and output (y or f(x)) values in two columns
Identify patterns in the table to determine the function type
Graphs plot points (x, y) on a coordinate plane and connect them to create a curve or line
Analyze the graph's key features to identify the function type
Equations express the relationship between the input and output using mathematical symbols
Identify the function type based on the equation's form (linear, quadratic, exponential)
Function operations and composition
Arithmetic operations on functions
Addition (f+g)(x)=f(x)+g(x)
Subtraction (f−g)(x)=f(x)−g(x)
Multiplication (f⋅g)(x)=f(x)⋅g(x)
Division (gf)(x)=g(x)f(x) where g(x)=0
of functions combines two or more functions by using the output of one as the input of another
(f∘g)(x)=f(g(x)): First, evaluate g(x), then use the result as the input for f
Domain of the is the domain of g restricted to values that produce outputs within the domain of f
Symmetry in functions
Even functions are symmetric about the y-axis
Definition: f(−x)=f(x) for all x in the domain
Examples: f(x)=x2, f(x)=cos(x)
Odd functions are symmetric about the origin
Definition: f(−x)=−f(x) for all x in the domain
Examples: f(x)=x3, f(x)=sin(x)
Functions that are neither even nor odd have no symmetry or are symmetric about other lines (e.g., y=x)
Example: f(x)=x2+x
Implications on graphs
Even functions: If (a,b) is on the graph, then (−a,b) is also on the graph
Odd functions: If (a,b) is on the graph, then (−a,−b) is also on the graph
Function Transformations and Continuity
alter the
Vertical shifts: f(x)+k moves the graph up k units
Horizontal shifts: f(x−h) moves the graph right h units
Vertical stretches/compressions: af(x) stretches (∣a∣>1) or compresses (0<∣a∣<1) vertically
Horizontal stretches/compressions: f(bx) stretches (0<∣b∣<1) or compresses (∣b∣>1) horizontally
Reflections: −f(x) reflects over the x-axis, f(−x) reflects over the y-axis
have no breaks, holes, or jumps in their graphs
A function f(x) is continuous at a point a if:
f(a) is defined
limx→af(x) exists
limx→af(x)=f(a)
A function is continuous on an interval if it is continuous at every point in that interval
have at least one point where continuity conditions are not met
Types of discontinuities:
Removable (point) discontinuity: A hole in the graph
Jump discontinuity: The function "jumps" from one value to another
Infinite discontinuity: The function approaches infinity as x approaches a certain value
"undo" the original function, swapping input and output
For a function f, its inverse f−1 satisfies f(f−1(x))=x and f−1(f(x))=x
The graph of an inverse function is the reflection of the original function over the line y=x