```Question 249140
f(x) = sqrt(x+2)

g(x) = (2x+3)

you want:

f(g(2))

first find g(2)

in the equation of g(x) = (2x+3), you replace x with 2 to get:

g(2) = (2*2 + 3) = 4+3 = 7

in the equation of f(x) = sqrt(x+2), you replace x with g(x) to get:

f(g(x) = sqrt(g(x)+2)

since g(x) = 7, you replace g(x) with 7 to get:

f(7) = sqrt(7+2) = sqrt(9) = 3

the equation is the rules by which you relate the independent variable to the dependent variable.

the independent variable is the variable that does not depend on any rules to establish a value for it.

the dependent variable is the variable that depends on the rules of the equation to establish a value for it.

when x is a certain value, y has to be a certain value.

x is also called the argument of the function.

the argument of the function is the variable that the rules of the equation are working on.

y = f(x) = x^2 means the independent variable in the equation is x.

y = f(a) = a^2 means the independent variable in the equation is a.

the rules are the same.

the only difference is the variable that the equation is working with.

that is why when you say f(x) = x^2, f(x) becomes the dependent variable and x is the independent variable.

when you say f(2) using the same equation, then the rules stay the same but the independent variable becomes 2 instead of x.

f(2) = 2^2 = 4.

when you say f(g(x)), then the independent variable in the equation becomes g(x).

g(x), however, was the dependent variable in another equation, so it's value had to be calculated before you could put it into f(g(x)).

that's exactly what we did above.

we solved for g(2) to get 7.

then we solved for f(g(x)) to get f(7) = 3.

you are placing g(x) into the argument of the equation f(x) = sqrt(x+2) which means you are replacing the existing argument of x with the argument of g(x).

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