The next concept to cover is that of the composite function, that is, a 'function of a function'.
Worked Examples
The two functions f and g are defined by
f : x → 4x − 3
g : x → 2x − 1
Find f(g(x)) and g(f(x)),
Show me| f(g(x)) | = 4(g(x)) − 3 | since, for example, f(a) = 4a − 3, etc. |
| = 4(2x − 1) − 3 | ||
| = 8x − 4 − 3 | ||
| = 8x − 7 |
| g(f(x)) | = 2(f(x)) − 1 |
| = 2(4x − 3) − 1 | |
| = 8x − 6 − 1 | |
| = 8x − 7 |
What are the values of f(g(0)) and g(f(0))?
Show mef(g(0)) = 8 × 0 − 7 = − 7
g(f(0)) = 8 × 0 − 7 = − 7
Are there any solutions of f(g(x)) = g(f(x)) ?
Show meAs f(g(x)) = g(f(x)), then this is satisfied by all values of x.
Note
The result f(g(x)) = gf(x) is not generally true, but depends on the values of the coefficients in the functions (see Question 5 in Exercises).
Note
We usually write, for short,
f(g(x)) = fg(x)
and
g(f(x)) = gf(x)
but you must remember what this notation means.
The functions f and g are defined by
f : x → x2 + 1
g : x → x − 1
Will fg(x) = gf(x) for any value of x ?
| fg(x) = f(g(x)) | = (g(x))2 + 1 | |
| = (x − 1)2 + 1 | ||
| = x2 − 2x + 1 + 1 | ||
| = x2 − 2x + 2 |
| gf(x) = g(f(x)) | = f(x) − 1 |
| = x2 + 1 − 1 | |
| = x2 |
If fg(x) = gf(x), then
| x2 − 2x + 2 | = x2 |
| −2x + 2 | = 0 |
| 2x | = 2 |
| x | = 1 |
So fg(x) = gf(x) when x = 1.
Exercises
