To find the functions f(x) and g(x) such that the function can be described as y = f(g(x)), given y = 102x + 9, we can take the following approach:
First, let’s define g(x) in a simple linear form. A common choice is to let g(x) = ax + b, where a and b are constants that we need to determine.
Next, we need to express y in terms of g(x). In this case, we want to find f such that:
y = f(g(x)) = f(ax + b)
Now, we need to choose f in a way that when we substitute g(x) into it, we get our original equation:
y = 102x + 9
Let’s now express y in terms of a transformation of g(x). One way to do this is to isolate x from g(x). For example, if we make a simple choice of:
g(x) = x
This means we are not changing x at all, which leads to:
f(g(x)) = f(x) = 102x + 9
Thus, we can define f(x) directly as:
f(x) = 102x + 9
So one set of solutions is:
- f(x) = 102x + 9
- g(x) = x
Alternatively, if we want to explore other combinations, we could choose different forms for g(x)—for example, we could let:
g(x) = 102x and then find f such that f(g(x)) = 102x + 9.
In this case, we would have to set:
f(g(x)) = g(x) + 9
This leads us to:
- f(x) = x + 9
- g(x) = 102x
In conclusion, there are multiple pairs of functions f(x) and g(x) that satisfy the original equation. The simplest solutions are:
- f(x) = 102x + 9, g(x) = x
Or more generally:
- f(x) = x + 9, g(x) = 102x
Both of these pairs provide valid descriptions of the function y = f(g(x)).