Composition of Functions
The composition of functions, denoted as (fog)(x) = f(g(x)), allows us to plug g(x) into f(x) as follows:
- When (fog)(x) is calculated, g(x) is plugged into f(x).
- When (gof)(x) is calculated, f(x) is plugged into g(x).
Example 1
Exploring the composition of functions with specific values:
- Given f(x) = 3x and g(x) = x - 5, we have:
- (fog)(x) = f(g(x)) = 3(x-5) = 3x -15
- (gof)(x) = g(f(x)) = 3x - 5
- (fog)(2) = 3·2-15 = 6 -15 = -9
Example 2
Another example with f(x) = 5x+2 and g(x) = 3x - 4:
- (fog)(x) = f(3x-4) = 5(3x-4) + 2 = 15x-20+2 = 15x-18
- (gof)(x) = g(5x+2) =3 (5x+2) -4 = 15x+6-4 = 15x+2
- (fog)(2)= 15·2-18=30-18=12
Example 3
Lastly, for f(x) = 7x+1 and g(x) = 2x² - 9:
- (fog)(x) = 7(2x² - 9) + 1 = 14x²-63+1 = 14x²-62
- (gof)(x) = 2(7x + 1)² -9 = 98x²+28x-7
- (gof)(2) = 14-2² -62 = 56-62=-6
Inverse Functions
The inverse function undoes what the original function does, where x and y swap places. If (a,b) is on the graph of a function, then (b,a) is on the graph of its inverse. The symbol f^(-1)(x) represents the inverse function of f(x).
Example
Considering f(x)=8x-19, the graph and its inverse can be obtained as follows:
- Original function equation: y=8x-19
- Swap x and y: x=8y-19
- Solve for y: x+19=8y
- Replace y with f^(-1)(x): f^(-1)(x) = x + 19 / 8
Inverse Function Calculation
We can determine f(x) if we know f^(-1)(x) by using the following steps:
- Step 1: Replace f(x) with y, e.g., y = 5x
- Step 2: Switch x and y, then solve for y
- Step 3: Replace y with f^(-1)(x)
Using the points (-3,0), (-1,2), and (5,3) on f(x), we can obtain their corresponding points on f^(-1)(x) as (0,-3), (2-1), and (3,5).
