Final answer:
The correct condition for f(g(x)) to equal g(f(x)) when given f(x)=ax+b and g(x)=cx+d is that b and d are equal and that ad=bc, which corresponds to option B, f(b)=g(b). This condition ensures that the compositions of f and g give equal results, which is what is needed for the functions to commute.
"The correct option is approximately option B"
Step-by-step explanation:
If f(x)=ax+b and g(x)=cx+d, then f(g(x))=g(f(x)) if and only if a certain condition is met among the values of the coefficients a, b, c, and d. To find the correct condition, we compose the functions f and g both ways and equate them.
First, let's compute f(g(x)):
f(g(x)) = f(cx+d) = a(cx+d) + b = acx + ad + b.
Now, let's compute g(f(x)):
g(f(x)) = g(ax+b) = c(ax+b) + d = cax + cb + d.
For f(g(x)) to equal g(f(x)), their expressions must be identical:
- acx + ad + b = cax + cb + d
The coefficients of x are already equal (ac = ca), so we are left with two equations:
Since option B suggests f(b) = g(b), we should verify this by plugging b into both f(x) and g(x):
- f(b) = a∙b + b
- g(b) = c∙b + d
And for f(b) to be equal to g(b), we require b+d = ab+b, which simplifies to d = ab. Since we also obtained the relation b = d from comparing f(g(x)) and g(f(x)), this leads to the conclusion that option B is the correct answer.