Question

Let f(x)=(1)/(x^(2)),g(x)=x+3 Find the composite function which expresses the given correspondence correctly. (1)/(x^(2)+6x+9)

Answer 1

The composite function is obtained by plugging the expression for g(x) into f(x).

To find the composite function, substitute g(x) into f(x):

f(g(x)) = f(x+3)

Now, let's plug this expression into f(x) and simplify:

f(g(x)) = f(x+3) = 1/((x+3)^(2))

So, the composite function that expresses the given correspondence correctly is 1/((x+3)^(2)).

This composite function represents a function that takes an input value, adds 3 to it, squares the sum, and then takes the reciprocal of the squared value.

For example, if we evaluate this composite function at x = 2:

f(g(2)) = 1/((2+3)^(2)) = 1/25

Therefore, the composite function is 1/((x+3)^(2)).

This response is approximately 170 words.