Education.com
Try
Brainzy
Try
Plus

Combinations of Functions Help

based on 4 ratings
By — McGraw-Hill Professional
Updated on Oct 4, 2011

Introduction to Combinations of Functions

Most of the functions studied in calculus are some combination of only a few families of functions, most of the combinations are arithmetic. We can add two functions, f + g(x), subtract them, fg (x), multiply them, fg(x), and divide them Combinations of Functions and Inverse Functions. The domain of f + g ( x ), fg(x), and fg(x), is the intersection of the domain of f(x) and g(x). In other words, their domain is where the domain of f(x) overlaps the domain of g(x). The domain of Combinations of Functions and Inverse Functions is the same, except we need to remove any x that makes g(x) = 0.

Examples

Find f + g(x), fg(x), fg(x), and Combinations of Functions and Inverse Functions Examples and their domain.

 

  • f ( x ) = x 2 − 2 x + 5 and g (x) = 6 x −10

    f + g (x) = f(x) + g (x) = (x2 − 2 x + 5) + (6 x − 10) = x 2 + 4 x − 5

    f - g (x) = f(x) − g (x) = (x 2 − 2 x + 5) − (6 x − 10) = x 2 − 8x + 15

    fg (x) = f(x) g (x) = (x2 − 2 x + 5)(6 x − 10) = 6 x 3 − 10 x2 − 12 x 2 + 20 x + 30 x − 50

    = 6 x 3 − 22 x2 + 50 x − 50

    Combinations of Functions and Inverse Functions Examples

    The domain of f + g(x), fg (x), and fg(x) is (−∞, ∞). The domain of Combinations of Functions and Inverse Functions Examples is Combinations of Functions and Inverse Functions Examples (from 6 x − 10 = 0), or Combinations of Functions and Inverse Functions Examples.

  • f ( x ) = x − 3 and Combinations of Functions and Inverse Functions Examples

    Combinations of Functions and Inverse Functions Examples

     

    The domain for f + g(x), fg (x), and fg(x) is [−2, ∞) (from x + 2 ≥ 0). The domain for Combinations of Functions and Inverse Functions Examples is (−2, ∞) because we need Combinations of Functions and Inverse Functions Examples.

Function Composition

An important combination of two functions is function composition . This involves evaluating one function at the other. The notation for composing f with g is f o g(x). By definition, f o g(x) = f (g(x)), this means that we substitute g(x) for x in f(x).

Examples

Find f o g(x) and g o f(x).

  • f (x) = x 2 + 1 and g (x) = 3 x + 2

    f o g(x) = f(g(x))

          = f (3 x + 2) Replace g (x) with 3 x + 2.

          = (3 x + 2) 2 + 1 Substitute 3 x + 2 for x in f(x).

          = (3x + 2)(3x + 2) + 1 = 9 x2 + 12 x + 5

    g o f ( x ) = g (f (x))

          = g(x2 + 1) Replace f(x) with x2 + 1.

          = 3( x 2 + 1) + 2 Substitute x 2 + 1 for x in g(x).

          = 3 x 2 + 3 + 2 = 3 x 2 + 5

  • Combinations of Functions and Inverse Functions Examples

    Combinations of Functions and Inverse Functions Examples

  • Combinations of Functions and Inverse Functions Examples

    Combinations of Functions and Inverse Functions Examples 

                          Combinations of Functions and Inverse Functions Examples

    Combinations of Functions and Inverse Functions Examples

At times, we only need to find f o g(x) for a particular value of x. The y-value for g(x) becomes the x -value for f(x).

Example

  • Find f o g (−1), f o g (0), and g o f (1) for f ( x ) = 4 x + 3 and g ( x ) = 2−x2.
  • f o g (−1) = f ( g (−1)) Compute g (−1).

          = f (1) g (−1) = 2−(−1) 2 = 1

          = 4(1) + 3 = 7 Evaluate f ( x ) at x = 1.

    f o g (0) = f ( g (0)) Compute g (0).

          = f (2) g (0) = 2 − 0 2 = 2

          = 4(2) + 3 = 11 Evaluate f(x) at x = 2.

    g o f (1) = g ( f (1)) Compute f (1).

          = g (7) f (1) =4(1) + 3 = 7

          = 2 − 7 2 = −47 Evaluate g ( x ) at x = 7.

We can compose two functions at a single x -value by looking at the graphs of the individual functions. To find f o g(a), we will look at the graph of g(x) to find the point whose x -coordinate is a. The y -coordinate of this point will be g(a). Then we will look at the graph of f(x) to find the point whose x -coordinate is g(a). The y -coordinate of this point will be f (g(a)) = f o g (a).

View Full Article
Add your own comment