6 1 Skills Practice Operations On Functions

6.1 Skills Practice: Operations on Functions

This comprehensive guide delves into the intricacies of operations on functions, a crucial concept in algebra and precalculus. We'll explore six key operations – addition, subtraction, multiplication, division, composition, and finding the inverse – providing detailed explanations, practical examples, and helpful tips to master these skills. Understanding these operations is foundational for more advanced mathematical concepts. Let's dive in!

1. Addition of Functions

The addition of functions is a straightforward operation. Given two functions, f(x) and g(x), their sum, denoted as (f + g)(x), is simply the sum of their individual outputs for the same input x.

Formula: (f + g)(x) = f(x) + g(x)

Example:

Let f(x) = 2x + 1 and g(x) = x² - 3. Find (f + g)(x).

(f + g)(x) = f(x) + g(x) = (2x + 1) + (x² - 3) = x² + 2x - 2

This means that for any value of x, the output of (f + g)(x) is the sum of the outputs of f(x) and g(x) for that same x value.

Domain of (f + g)(x): The domain of the sum of two functions is the intersection of the domains of the individual functions. In simpler terms, it includes only the x values for which both f(x) and g(x) are defined.

2. Subtraction of Functions

Similar to addition, the subtraction of functions involves finding the difference between the outputs of two functions for the same input x.

Formula: (f - g)(x) = f(x) - g(x)

Example:

Using the same functions from the previous example, f(x) = 2x + 1 and g(x) = x² - 3, let's find (f - g)(x).

(f - g)(x) = f(x) - g(x) = (2x + 1) - (x² - 3) = -x² + 2x + 4

Again, note the order of operations: we subtract g(x) from f(x).

Domain of (f - g)(x): Like addition, the domain of the difference of two functions is the intersection of their individual domains.

3. Multiplication of Functions

The multiplication of functions involves multiplying the outputs of two functions for the same input x.

Formula: (f * g)(x) = f(x) * g(x)

Example:

Using our familiar functions, f(x) = 2x + 1 and g(x) = x² - 3, let's find (f * g)(x).

(f * g)(x) = f(x) * g(x) = (2x + 1)(x² - 3) = 2x³ - 6x + x² - 3 = 2x³ + x² - 6x - 3

Domain of (f * g)(x): The domain of the product of two functions is also the intersection of their individual domains.

4. Division of Functions

The division of functions involves dividing the output of one function by the output of another for the same input x.

Formula: (f / g)(x) = f(x) / g(x)

Example:

With f(x) = 2x + 1 and g(x) = x² - 3, let's find (f / g)(x).

(f / g)(x) = f(x) / g(x) = (2x + 1) / (x² - 3)

Domain of (f / g)(x): This is where things get slightly more complex. The domain of the quotient of two functions is the intersection of their individual domains, excluding any values of x that make the denominator g(x) equal to zero. In this example, we must exclude the values of x that make x² - 3 = 0, which are x = √3 and x = -√3.

5. Composition of Functions

Composition of functions is a more sophisticated operation. It involves applying one function to the output of another function. This is often represented as (f ∘ g)(x) or f(g(x)).

Formula: (f ∘ g)(x) = f(g(x))

Example:

Let's use f(x) = 2x + 1 and g(x) = x² - 3. Let's find (f ∘ g)(x) and (g ∘ f)(x).

(f ∘ g)(x) = f(g(x)) = f(x² - 3) = 2(x² - 3) + 1 = 2x² - 5

(g ∘ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)² - 3 = 4x² + 4x + 1 - 3 = 4x² + 4x - 2

Notice that (f ∘ g)(x) ≠ (g ∘ f)(x). The order of composition matters!

Domain of (f ∘ g)(x): The domain of a composite function is determined by considering the domain of the inner function (g(x) in this case) and ensuring that the output of the inner function is within the domain of the outer function (f(x)).

6. Finding the Inverse of a Function

The inverse of a function, denoted as f⁻¹(x), "undoes" the operation of the original function. If f(a) = b, then f⁻¹(b) = a. Not all functions have inverses; a function must be one-to-one (each input maps to a unique output) to have an inverse.

Finding the Inverse:

To find the inverse of a function, follow these steps:

1. Replace f(x) with y: y = f(x)
2. Swap x and y: x = f(y)
3. Solve for y: This step often involves algebraic manipulation.
4. Replace y with f⁻¹(x): The resulting expression is the inverse function.

Example:

Let's find the inverse of f(x) = 3x - 6.

1. y = 3x - 6
2. x = 3y - 6
3. x + 6 = 3y
4. y = (x + 6) / 3
5. Therefore, f⁻¹(x) = (x + 6) / 3

Domain and Range: The domain of a function is the range of its inverse, and the range of a function is the domain of its inverse. This is an important relationship to remember.

Advanced Considerations and Practice Problems

Mastering these six operations requires consistent practice. Here are some advanced considerations and practice problems to solidify your understanding:

1. Piecewise Functions: Operations on piecewise functions require careful consideration of the different pieces of the function and their respective domains.

2. Trigonometric Functions: Applying these operations to trigonometric functions introduces the complexities of periodic functions and their domains.

3. Graphing: Graphing the results of these operations can provide valuable visual insights into the transformations and relationships between functions. Software like Desmos or GeoGebra can be extremely helpful for visualization.

4. Real-World Applications: These operations are fundamental to many real-world applications in fields like physics, engineering, and economics, where functions model various phenomena.

Practice Problems:

1. Given f(x) = x² + 2x and g(x) = x - 3, find:

   • (f + g)(x)
   • (f - g)(x)
   • (f * g)(x)
   • (f / g)(x) (State the domain)
   • (f ∘ g)(x)
   • (g ∘ f)(x)

2. Find the inverse of the following functions:

   • h(x) = 5x + 10
   • k(x) = √(x - 4) (State the domain and range of both the function and its inverse)
   • m(x) = (x + 2)² (Is there an inverse for the entire function? Why or why not?)

3. Let f(x) = |x| and g(x) = x + 1. Find (f ∘ g)(x) and (g ∘ f)(x) and sketch their graphs. How do the graphs illustrate the effects of composition?

4. Given f(x) = 2x + 1 and g(x) = x / (x -2). Find the domain of (f/g)(x)

By working through these problems and exploring the advanced considerations, you'll develop a strong foundation in operations on functions. Remember that consistent practice is key to mastering these essential mathematical skills. Good luck!