Functions Test ...

$$f(x) = x^{2} + d$$ and $$g(x) = 2x^{2}$$, where d is a constant. If $$\dfrac {f(g(2))}{f(2)} = 4$$, find the value of $$d$$.

If $$f(x) =$$ $$x^2$$ and $$g(x) = 2x$$, calculate the value of $$f(g(-3))-g(f(-3))$$.

Find the correct expression for $$\displaystyle f\left( g\left( x \right)  \right) $$ if $$\displaystyle f(x)=4x+1$$ and $$\displaystyle g\left( x \right) ={ x }^{ 2 }-2$$  

If $$p(x) = \dfrac{x}{x-2}$$ and $$q(x) = \sqrt{9-x}$$, find the value of $$(p\circ  q)(5)$$

Consider the functions $$\displaystyle f\left( x \right) =\sqrt { x } $$ and $$\displaystyle g\left( x \right) =7x+b$$. Find the value of $$b$$, if the composite function, $$\displaystyle y=f\left( g\left( x \right)  \right) $$ passes through $$(4, 6)$$. 

Find $$g(x)$$, if $$f(x) = 7x + 12$$ and $$f(g(x) = 21x^{2} + 40$$

Given a function $$f(x) = \dfrac {1}{2}x - 4$$ and the composite function $$f(g(x)) = g(f(x))$$, determine which among the following can be $$g(x)$$:
I. $$2x - \dfrac {1}{4}$$
II. $$2x + 8$$
III. $$\dfrac {1}{2}x - 4$$

If $$f(x) = 2x$$ and $$f(f(x)) = x + 1$$, then the value of $$x $$ is

Find the value of  $$g(f(2))$$, if $$f(x) = e^{x}$$ and $$g(x) = \dfrac {x}{2}$$

The above figure shows the graph of the function $$f(x)$$, the value of $$f(f(3))$$ is: