To find f(g(x)), we first need to understand what each function represents.
The function f(x) is given as:
f(x) = 8x³ + 28x + 61
And the function g(x) is:
g(x) = 2x + 5
Now, to find f(g(x)), we will substitute g(x) into the function f. This means wherever we see x in f(x), we will replace it with g(x).
Thus, we have:
f(g(x)) = f(2x + 5)
Now, we replace x in f(x):
f(2x + 5) = 8(2x + 5)³ + 28(2x + 5) + 61
This requires us to compute (2x + 5)³:
(2x + 5)³ = (2x + 5)(2x + 5)(2x + 5)
Using the binomial theorem, we can expand (2x + 5)³:
(2x)³ + 3(2x)²(5) + 3(2x)(5)² + (5)³ = 8x³ + 60x² + 150x + 125
Now substituting (2x + 5)³ back into f:
f(2x + 5) = 8(8x³ + 60x² + 150x + 125) + 28(2x + 5) + 61
Calculating this step-by-step:
8(8x³ + 60x² + 150x + 125) = 64x³ + 480x² + 1200x + 1000
Next:
28(2x + 5) = 56x + 140
So, summing it all together:
f(2x + 5) = (64x³ + 480x² + 1200x + 1000) + (56x + 140) + 61
Combining like terms:
f(2x + 5) = 64x³ + 480x² + (1200x + 56x) + (1000 + 140 + 61)
This gives us:
f(2x + 5) = 64x³ + 480x² + 1256x + 1201
Therefore, the result of f(g(x)) is:
f(g(x)) = 64x³ + 480x² + 1256x + 1201