To find the composition, \(\displaystyle f\circ g\) given

\(\displaystyle \\f(x)=x^2+2x \\g(x)=3x\)

recall what a composition of two functions represents.

\(\displaystyle f \circ g=f(g(x))\)

This means that the function \(\displaystyle g(x)\) will replace each \(\displaystyle x\) in the function \(\displaystyle f(x)\).

Therefore, in this particular problem the composition becomes

\(\displaystyle \\f\circ g=f(g(x)) \\f(g(x))=(3x)^2+2(3x) \\f(g(x))=9x^2+6x\)

To find the inverse of a function swap the variables and solve for \(\displaystyle y\). The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function. 

For this particular function the inverse is found as follows.

\(\displaystyle f(x)=\frac{3}{2+5x}\)

\(\displaystyle y=\frac{3}{2+5x}\)

First, switch the variables.

\(\displaystyle x=\frac{3}{2+5y}\)

Now, perform algebraic operations to solve for \(\displaystyle y\).

\(\displaystyle x\cdot (2+5y)={3}\)

\(\displaystyle \frac{x\cdot(2+5y)}{x}=\frac{3}{x}\)

\(\displaystyle \\2+5y=\frac{3}{x} \\\\5y=\frac{3}{x}-2 \\\\y=\frac{3}{5x}-\frac{2}{5}\)

Therefore, the inverse is

\(\displaystyle f^{-1}(x)=\frac{3}{5x}-\frac{2}{5}\)

To find the composition, \(\displaystyle f\circ g\) given

\(\displaystyle \\f(x)=2x^2+3 \\g(x)=7x\)

recall what a composition of two functions represents.

\(\displaystyle f \circ g=f(g(x))\)

This means that the function \(\displaystyle g(x)\) will replace each \(\displaystyle x\) in the function \(\displaystyle f(x)\).

Therefore, in this particular problem the composition becomes

\(\displaystyle \\f\circ g=f(g(x)) \\f(g(x))=2(7x)^2+3 \\f(g(x))=2(49x^2)+3\\f(g(x))=98x^2+3\)

To find the inverse of a function swap the variables and solve for \(\displaystyle y\). The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function. 

For this particular function the inverse is found as follows.

\(\displaystyle f(x)=2x^2+4\)

\(\displaystyle y=2x^2+4\)

First, switch the variables.

\(\displaystyle x=2y^2+4\)

Now, perform algebraic operations to solve for \(\displaystyle y\).

\(\displaystyle \\x-4=2y^2 \\\\\frac{x-4}{2}=y^2 \\\\\frac{x}{2}-2=y^2 \\\\\sqrt{\frac{x}{2}-2}=y\)

Therefore, the inverse is

\(\displaystyle f^{-1}(x)=\sqrt{\frac{x}{2}-2}\)

To find the composition, \(\displaystyle f\circ g\) given

\(\displaystyle \\f(x)=x^2 \\\\g(x)=\frac{1}{2}x\)

recall what a composition of two functions represents.

\(\displaystyle f \circ g=f(g(x))\)

This means that the function \(\displaystyle g(x)\) will replace each \(\displaystyle x\) in the function \(\displaystyle f(x)\).

Therefore, in this particular problem the composition becomes

\(\displaystyle \\f\circ g=f(g(x)) \\\\f(g(x))=\left(\frac{1}{2}x \right )^2\\\\f(g(x))=\frac{1}{4}x^2\)

To find the composition, \(\displaystyle f\circ g\) given

\(\displaystyle \\f(x)=x^2+2x \\g(x)=3x\)

recall what a composition of two functions represents.

\(\displaystyle f \circ g=f(g(x))\)

This means that the function \(\displaystyle g(x)\) will replace each \(\displaystyle x\) in the function \(\displaystyle f(x)\).

Therefore, in this particular problem the composition becomes

\(\displaystyle \\f\circ g=f(g(x)) \\f(g(x))=(3x)^2+2(3x) \\f(g(x))=9x^2+6x\)

To find the inverse of a function swap the variables and solve for \(\displaystyle y\). The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function. 

For this particular function the inverse is found as follows.

\(\displaystyle f(x)=\frac{3}{2+5x}\)

\(\displaystyle y=\frac{3}{2+5x}\)

First, switch the variables.

\(\displaystyle x=\frac{3}{2+5y}\)

Now, perform algebraic operations to solve for \(\displaystyle y\).

\(\displaystyle x\cdot (2+5y)={3}\)

\(\displaystyle \frac{x\cdot(2+5y)}{x}=\frac{3}{x}\)

\(\displaystyle \\2+5y=\frac{3}{x} \\\\5y=\frac{3}{x}-2 \\\\y=\frac{3}{5x}-\frac{2}{5}\)

Therefore, the inverse is

\(\displaystyle f^{-1}(x)=\frac{3}{5x}-\frac{2}{5}\)

To find the composition, \(\displaystyle f\circ g\) given

\(\displaystyle \\f(x)=2x^2+3 \\g(x)=7x\)

recall what a composition of two functions represents.

\(\displaystyle f \circ g=f(g(x))\)

This means that the function \(\displaystyle g(x)\) will replace each \(\displaystyle x\) in the function \(\displaystyle f(x)\).

Therefore, in this particular problem the composition becomes

\(\displaystyle \\f\circ g=f(g(x)) \\f(g(x))=2(7x)^2+3 \\f(g(x))=2(49x^2)+3\\f(g(x))=98x^2+3\)

To find the inverse of a function swap the variables and solve for \(\displaystyle y\). The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function. 

For this particular function the inverse is found as follows.

\(\displaystyle f(x)=2x^2+4\)

\(\displaystyle y=2x^2+4\)

First, switch the variables.

\(\displaystyle x=2y^2+4\)

Now, perform algebraic operations to solve for \(\displaystyle y\).

\(\displaystyle \\x-4=2y^2 \\\\\frac{x-4}{2}=y^2 \\\\\frac{x}{2}-2=y^2 \\\\\sqrt{\frac{x}{2}-2}=y\)

Therefore, the inverse is

\(\displaystyle f^{-1}(x)=\sqrt{\frac{x}{2}-2}\)

To simplify the radical expression look at the factors under each radical.

\(\displaystyle \sqrt{125}+\sqrt{48}\)

\(\displaystyle \sqrt{25\cdot 5}+\sqrt{4\cdot 12}\)

Recall that 25 and 4 are perfect squares.

\(\displaystyle 5\sqrt{5}+2\sqrt{12}\)

From here 12 can be factored further.

\(\displaystyle \\5\sqrt{5}+2\sqrt{4\cdot 3} \\5\sqrt{5}+2\cdot 2\sqrt{3} \\5\sqrt{5}+4\sqrt{3}\)