Functions, f and g are given by f(x)= 3+ cos x and g(x) = 2x, x is a real number. Determine the value of c for which f(g(x))= g(f(x)) where 0 x<2

Step-by-step explanation:

Given two functions, we can combine them in such a way so that the outputs of one function become the inputs of the other. This action defines a composite function. Let's take a look at what this means!

Evaluating composite functions

Example

If f(x)=3x-1f(x)=3x−1f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1 and g(x)=x^3+2g(x)=x

3

+2g, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2, then what is f(g(3))f(g(3))f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis?

Solution

One way to evaluate f(g(3))f(g(3))f, left parenthesis, g, left parenthesis, 3, right parenthesis, right parenthesis is to work from the "inside out". In other words, let's evaluate g(3)g(3)g, left parenthesis, 3, right parenthesis first and then substitute that result into fff to find our answer.

Let's evaluate g({3})g(3)g, left parenthesis, 3, right parenthesis.

\begin{aligned}g(x)&=x^3+2 g(3)&=({3})^3 +2\small{\gray{\text{Plug in }x={3.}}} &={29}\end{aligned}

g(x)

g(3)

​

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!

a)(x-2)^2+(y-1)^2=9

step-by-step explanation:

(picture explains)