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Function Operations

Addition, Subtraction, Multiplication, and Division

• Function operations involve combining two functions, denoted as f(x) and g(x), to generate new functions that can model relationships or transformations.

• Addition: The sum of two functions, represented as $(f + g)(x) = f(x) + g(x)$, combines their output values at each point in the domain. This operation can be useful in various applications, such as computing total sales from two different product lines.

• Subtraction: The difference of two functions, represented as $(f - g)(x) = f(x) - g(x)$, provides the difference in their outputs at each x-value. This can be useful in scenarios such as calculating profit by subtracting costs from revenues.

• Multiplication: The product of two functions, denoted as $(f \times g)(x) = f(x) \times g(x)$, results in a new function that represents the product of the values from f and g at each x-value. This operation is common in areas such as physics, where you might calculate force by multiplying mass and acceleration.

• Division: The quotient of two functions, expressed as $(f / g)(x) = \frac{f(x)}{g(x)}$, gives the ratio of their outputs. It is important to consider the restrictions on the domain since this operation cannot include undefined values where g(x) equals zero.

• Domain Considerations: The domain of the sum, difference, product, and quotient functions consists of the x-values that are within the domains of both f(x) and g(x). If either function is undefined at a particular x-value, that value must be excluded from the resulting function's domain. Notably, the domain of the quotient function requires that g(x) is not equal to zero to avoid indeterminate forms.

Example 1

• Given:
    - $f(x) = 3x + 8$
    - $g(x) = 2x - 12$

• Find the results of:
    - $f + g$
    - $f - g$
    - $f \times g$
    - $f / g$

• State the domain of each resulting function.

Example 2

• Given:
    - $f(x) = x^2 - 1$
    - $g(x) = x + 1$

Composition of Functions

• Definition: Composition of functions is defined as applying one function to the result of another function, denoted as $(f \circ g)(x) = f(g(x))$, which represents a nested operation that often arises in mathematical modeling.

• Important Note: Always adhere to the order of operations by starting with the inside function and working your way out to avoid errors in evaluation, especially in complex functions.

Example 3

• Given:
    - $f(x) = x^3$
    - $g(x) = x^2$

• Find:
    - $f(2)$
    - $g(2)$
    - $f(g(2))$

Example 4

• Given:
    - $f(x) = x - 2$
    - $g(x) = x^2$

• Find the following:
    - $(f \circ g)(2)$
    - $(g \circ f)(-5)$
    - $(f \circ g)(x)$

Inverse Functions

• Definition: An inverse function essentially reverses the effect of the original function. If the point (a, b) is on the function, then (b, a) is on the inverse function, indicating the swap of inputs and outputs. Graphically, every inverse function is a reflection over the line $y = x$, demonstrating the symmetry of their relationships.

• For example, let:
    - $f(x) = 2x + 1$
    - To find its inverse function, represented as $f^{-1}(x)$, follow the algebraic process described below.

Steps for Finding Inverses Algebraically

1. Replace $f(x)$ with y to simplify notation.

2. Interchange x and y, effectively switching the roles of input and output.

3. Solve for y to express it in terms of x. This new equation will be the inverse function.

4. Rewrite the resulting expression as $f^{-1}(x)$ to denote it as the inverse.

Example Problems on Inverse Functions

Example 6

• Given:
    - $y = x^2 + 3$

• Tasks:
    - Graph the function.
    - Determine if it is a function by checking whether it passes the vertical line test.
    - Algebraically find the inverse and graph it.
    - Determine if the inverse is also a function by assessing whether it passes the vertical line test.

Example 7

• Given:
    - $f(x) = x + 1$

• Tasks:
    - Graph the function.
    - Check if it is a function.
    - Algebraically find the inverse and graph it.
    - Determine if the inverse is a function.

Example 8

• Given:
    - Find the inverse of the function $f(x) = 2^{x + 3}$ by following the outlined steps for finding inverses.

Example 9

• Given:
    - Find the inverse of the function $f(x) = \frac{x + 2}{x - 4}$ following the proper algebraic techniques.

Example 10

• Task: Determine if the following functions are inverses of each other by checking if applying one function to the other yields the identity function.
    - a)
    - b)