Mastering how to combine functions is really important in Algebra II. It helps us understand how different equations relate to each other. Let's break down the four main operations: addition, subtraction, multiplication, and division.

1. Adding and Subtracting Functions

When you add or subtract two functions, you just work with their results. If we have two functions, ( f(x) ) and ( g(x) ), here’s how you find their sum and difference:

• Sum: ( (f + g)(x) = f(x) + g(x) )
• Difference: ( (f - g)(x) = f(x) - g(x) )

Example: If ( f(x) = 2x + 3 ) and ( g(x) = x^2 ), then:

• ( (f + g)(x) = (2x + 3) + (x^2) = x^2 + 2x + 3 )
• ( (f - g)(x) = (2x + 3) - (x^2) = -x^2 + 2x + 3 )

2. Multiplying Functions

Multiplying functions works in a similar way:

• Product: ( (f \cdot g)(x) = f(x) \cdot g(x) )

Example: Using the same functions:

• ( (f \cdot g)(x) = (2x + 3)(x^2) = 2x^3 + 3x^2 )

3. Dividing Functions

Dividing functions is also straightforward:

• Quotient: ( (f/g)(x) = \frac{f(x)}{g(x)} ), but remember ( g(x) ) can't be zero.

Example:

• ( (f / g)(x) = \frac{2x + 3}{x^2} )

4. Inverse Functions

Finding inverse functions is like reversing what the original function did. For a function ( f ), its inverse is shown as ( f^{-1} ), and it follows this rule: ( f(f^{-1}(x)) = x ).

Practice these ways of combining functions, and you’ll get the hang of it in no time!