Understanding Functions

• Functions are mathematical relationships where each input has exactly one output

Representing Functions

• Functions can be represented through equations, graphs, tables, and verbal descriptions

Function Notation

• Functions are commonly denoted as f(x), where 'f' names the function and 'x' represents the input
• f(x) is read as "f of x," and represents the output value of the function for a given input 'x'

Domain and Range

• The domain of a function is the set of all possible input values (x)
• The range is the set of all possible output values (f(x))

Function Evaluation

• To evaluate a function, substitute a specific value for 'x' into the function's equation and simplify
• For example, if f(x) = x^2 + 3, then f(2) = 2^2 + 3 = 7

Basic Function Operations

• Functions can be combined using arithmetic operations such as addition, subtraction, multiplication, and division

Addition of Functions

• (f + g)(x) = f(x) + g(x)
• To add two functions, add their corresponding output values for each 'x'

Subtraction of Functions

• (f - g)(x) = f(x) - g(x)
• To subtract two functions, subtract the output value of g(x) from f(x) for each 'x'

Multiplication of Functions

• (f * g)(x) = f(x) * g(x)
• To multiply two functions, multiply their output values for each 'x'

Division of Functions

• (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
• To divide two functions, divide the output value of f(x) by g(x) for each 'x', excluding any 'x' where g(x) = 0

Composition of Functions

• The composition of functions combines two functions by using the output of one function as the input for the other

Notation for Composition

• (f ∘ g)(x) = f(g(x))
• This is read as "f of g of x," meaning that the function 'g' is applied to 'x' first, and then 'f' is applied to the result

Evaluating Composite Functions

• First, evaluate the inner function, g(x), for a given 'x'
• Then, substitute the result from g(x) into the outer function, f(x)

Domain of Composite Functions

• The domain of (f ∘ g)(x) includes all 'x' in the domain of 'g' such that g(x) is in the domain of 'f'

Graphing Functions

• The graph of a function is a visual representation of the relationship between 'x' and f(x)

Coordinate Plane

• The graph is drawn on a coordinate plane, where the horizontal axis represents the input values (x) and the vertical axis represents the output values (f(x))

Plotting Points

• To graph a function, plot points (x, f(x)) on the coordinate plane
• Connect the points to form a curve or line representing the function

Common Types of Functions and Their Graphs

• Linear Functions: f(x) = mx + b (straight line)
• Quadratic Functions: f(x) = ax^2 + bx + c (parabola)
• Exponential Functions: f(x) = a^x (curved, increasing or decreasing rapidly)
• Absolute Value Functions: f(x) = |x| (V-shaped)

Transformations of Functions

• Transformations alter the position, size, or shape of a function's graph

Vertical Shifts

• Adding a constant 'c' to a function shifts the graph vertically
• f(x) + c shifts the graph upward by 'c' units
• f(x) - c shifts the graph downward by 'c' units

Horizontal Shifts

• Replacing 'x' with (x - c) shifts the graph horizontally
• f(x - c) shifts the graph to the right by 'c' units
• f(x + c) shifts the graph to the left by 'c' units

Vertical Stretching and Compression

• Multiplying a function by a constant 'a' stretches or compresses the graph vertically
• a * f(x) stretches the graph vertically if |a| > 1
• a * f(x) compresses the graph vertically if 0 < |a| < 1

Horizontal Stretching and Compression

• Replacing 'x' with (ax) stretches or compresses the graph horizontally
• f(ax) compresses the graph horizontally if |a| > 1
• f(ax) stretches the graph horizontally if 0 < |a| < 1

Reflections

• Reflections flip the graph across an axis
• -f(x) reflects the graph across the x-axis
• f(-x) reflects the graph across the y-axis