Question

Given f(x)=x^2+2x+3 and g(x)=x+4/3 solve for f(g(x)) when x=2

Answer 1

Answer:

$\displaystyle\mathsf{f(g(2)) \:=\:\frac{187}{9}}$

Step-by-step explanation:

We are provided with the following functions:

f(x) = x² + 2x + 3

$\displaystyle\mathsf{ g(x)\:=\:x+\frac{4}{3} }$

The given problem also requires to find the Composition of Functions, f(g(x)) when x = 2.

The Composition of Function f with function g can be expressed as ( f ° g )(x) = f(g(x)).  In solving for the composition of functions, we must first evaluate the innermost function, g(x), then use the output as an input for f(x).

Solve for f(g(x)) when x = 2:

Find g(x):

Starting with g(x), we will use x = 2 as an input value into the function:

$\displaystyle\mathsf{ g(x)\:=\:x+\frac{4}{3} }$

$\displaystyle\mathsf{ g(2)\:=\:(2)+\frac{4}{3} }$

Transform the first term, x = 2, into a fraction with a denominator of 3 to combine with 4/3:

$\displaystyle\mathsf{ g(2)\:=\:\frac{2\: \times\ 3}{3}+\frac{4}{3} }$

$\displaystyle\mathsf{ g(2)\:=\:\frac{6}{3}+\frac{4}{3}\:=\:\frac{6+4}{3}}$

$\displaystyle\mathsf{ g(2)\:=\:\frac{10}{3} }$

$\displaystyle\mathsf{Therefore,\:\: g(2)\:=\:\frac{10}{3} }$

Find f(x):

Next, we will use  $\displaystyle\mathsf{\frac{10}{3}}$ as input for the function, f(x) = x² + 2x + 3:

f(x) = x² + 2x + 3

$\displaystyle\mathsf{f\Bigg (\frac{10}{3}\Bigg)\:=\:x^2 \:+ 2x\:+\:3}$

$\displaystyle\mathsf{f\Bigg (\frac{10}{3}\Bigg) \:=\:\Bigg (\frac{10}{3}\Bigg)^{2}\:+ 2\Bigg(\frac{10}{3}\Bigg) \:+\:3}$

Use the Quotient-to-Power Rule of Exponents onto the leading term (x²):

$\displaystyle\mathsf{Quotient-to-Power\:\:Rule:\:\: \Bigg(\frac{a}{b}\Bigg)^m\:=\:\frac{a^m}{b^m} }$

$\displaystyle\mathsf{f\Bigg (\frac{10}{3}\Bigg) \:=\:\Bigg (\frac{10\:^2}{3\:^2}\Bigg)\:+ 2\Bigg(\frac{10}{3}\Bigg) \:+\:3}$

Multiply the numerator (10) of the middle term by 2:

$\displaystyle\mathsf{f\Bigg (\frac{10}{3}\Bigg) \:=\:\Bigg (\frac{100}{9}\Bigg)\:+ \Bigg(\frac{20}{3}\Bigg) \:+\:\frac{3}{1}}$

• Determine the least common multiple (LCM) of the denominators from the previous step: 9, 3, and 1 (which is 9).
• Then, transform the denominators of 20/3 and 3/1 on the right-hand side of the equation into like-fractions:

                       $\displaystyle\mathsf{\frac{20}{3}\Rightarrow \:\frac{20\:\times\ 3}{3\:\times\ 3} =\:\frac{60}{9}}$

                        $\displaystyle\mathsf{\frac{3}{1}\Rightarrow \:\frac{3\:\times\ 9}{1\:\times\ 9} =\:\frac{27}{9}}$

Finally, add the three fractions on the right-hand side of the equation:

$\displaystyle\mathsf{f\Bigg (\frac{10}{3}\Bigg) \:=\:\Bigg (\frac{100}{9}\Bigg)\:+ \Bigg(\frac{60}{9}\Bigg) \:+\:\frac{27}{9}\:=\:\frac{187}{9}}$

Final Answer:

$\displaystyle\mathsf{Therefore,\:\:f(g(2)) \:=\:\frac{187}{9}.}$

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Keywords:

Composition of functions

f o g

f (g(x))

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Learn more about Composition of Functions here:

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