Question

Find

(a) \({\bf{f + g}}\)

(b) \({\bf{f}} - {\bf{g}}\)

(c) \({\bf{fg}}\)

(d) \({\bf{f}}/{\bf{g}}\)

and state their domains.

37. \({\bf{f}}\left( x \right) = {x^3} + 2{x^2}\) \(g\left( x \right) = 3{x^2} - 1\)

Short answer

(a) The sum of the two functions is \(\left( {{\bf{f + g}}} \right)\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{3}}}{\bf{ + 5}}{{\bf{x}}^{\bf{2}}}{\bf{ - 1}}\) with domain \(\left( {{\bf{ - }}\infty {\bf{,}}\infty } \right)\).

(b) The difference of the two functions is \(\left( {{\bf{f - g}}} \right)\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{3}}}{\bf{ - }}{{\bf{x}}^{\bf{2}}}{\bf{ + 1}}\) with domain \(\left( {{\bf{ - }}\infty {\bf{,}}\infty } \right)\).

(c) The product of the two functions is \(\left( {{\bf{fg}}} \right)\left( {\bf{x}} \right){\bf{ = 3}}{{\bf{x}}^{\bf{5}}}{\bf{ + 6}}{{\bf{x}}^{\bf{4}}}{\bf{ - }}{{\bf{x}}^{\bf{3}}}{\bf{ - 2}}{{\bf{x}}^{\bf{2}}}\)with domain \(\left( {{\bf{ - }}\infty {\bf{,}}\infty } \right)\).

(d) The division of the two functions is \(\left( {{\bf{f/g}}} \right)\left( {\bf{x}} \right){\bf{ = }}\frac{{{{\bf{x}}^{\bf{3}}}{\bf{ + 2}}{{\bf{x}}^{\bf{2}}}}}{{{\bf{3}}{{\bf{x}}^{\bf{2}}}{\bf{ - 1}}}}\) with domain \(\left( {{\bf{ - }}\infty {\bf{,}}\infty } \right);x \ne \pm \sqrt {\frac{1}{3}} \).

Step-by-step solution

Given data

The provided functions are

\(f\left( x \right) = {x^3} + 2{x^2}\)

\(g\left( x \right) = 3{x^2} - 1\)

Combinations of functions

The combination of two functions is written as

\(\begin{array}{l}\left( {{\bf{f + g}}} \right)\left( {\bf{x}} \right){\bf{ = f}}\left( {\bf{x}} \right){\bf{ + g}}\left( {\bf{x}} \right)\;\;\;\;\;.....\left( {\bf{1}} \right)\\\left( {{\bf{f - g}}} \right)\left( {\bf{x}} \right){\bf{ = f}}\left( {\bf{x}} \right){\bf{ - g}}\left( {\bf{x}} \right)\;\;\;\;\;.....\left( {\bf{2}} \right)\\\left( {{\bf{fg}}} \right)\left( {\bf{x}} \right){\bf{ = f}}\left( {\bf{x}} \right){\bf{g}}\left( {\bf{x}} \right)\;\;\;\;\;.....\left( {\bf{3}} \right)\\\left( {{\bf{f/g}}} \right)\left( {\bf{x}} \right){\bf{ = f}}\left( {\bf{x}} \right){\bf{/g}}\left( {\bf{x}} \right)\;\;\;\;\;.....\left( {\bf{4}} \right)\end{array}\)

If the domain of \(f\left( x \right)\) is \(A\) and the domain of \(g\left( x \right)\) is B, then the domain of their combination is \({\bf{A}} \cap {\bf{B}}\) .

Sum of two functions

From equation (1),

\(\begin{array}{c}\left( {f + g} \right)\left( x \right) = {x^3} + 2{x^2} + 3{x^2} - 1\\ = {x^3} + 5{x^2} - 1\end{array}\)

Domain of \(f\left( x \right)\) is

\(A = \left( { - \infty ,\infty } \right)\)

Domain of \(g\left( x \right)\) is

\(B = \left( { - \infty ,\infty } \right)\)

Thus, domain of \(\left( {f + g} \right)\left( x \right)\) is

\(A \cap B = \left( { - \infty ,\infty } \right)\)

The combined function is \({x^3} + 5{x^2} - 1\) and the domain is \(\left( { - \infty ,\infty } \right)\).

Difference of two functions

From equation (2),

\(\begin{array}{c}\left( {f - g} \right)\left( x \right) = {x^3} + 2{x^2} - 3{x^2} + 1\\ = {x^3} - {x^2} + 1\end{array}\)

Domain of \(f\left( x \right)\) is

\(A = \left( { - \infty ,\infty } \right)\)

Domain of \(g\left( x \right)\) is

\(B = \left( { - \infty ,\infty } \right)\)

Thus, domain of \(\left( {f - g} \right)\left( x \right)\) is

\(A \cap B = \left( { - \infty ,\infty } \right)\)

The combined function is \({x^3} - {x^2} + 1\) and the domain is \(\left( { - \infty ,\infty } \right)\).

Product of two functions

From equation (3),

\(\begin{array}{c}\left( {fg} \right)\left( x \right) = \left( {{x^3} + 2{x^2}} \right)\left( {3{x^2} - 1} \right)\\ = 3{x^5} - {x^3} + 6{x^4} - 2{x^2}\\ = 3{x^5} + 6{x^4} - {x^3} - 2{x^2}\end{array}\)

Domain of \(f\left( x \right)\) is

\(A = \left( { - \infty ,\infty } \right)\)

Domain of \(g\left( x \right)\) is

\(B = \left( { - \infty ,\infty } \right)\)

Thus, domain of \(\left( {fg} \right)\left( x \right)\) is

\(A \cap B = \left( { - \infty ,\infty } \right)\)

The combined function is \(3{x^5} + 6{x^4} - {x^3} - 2{x^2}\) and the domain is \(\left( { - \infty ,\infty } \right)\).

Division of two functions

From equation (4),

\(\begin{array}{c}\left( {f/g} \right)\left( x \right) = \left( {{x^3} + 2{x^2}} \right)/\left( {3{x^2} - 1} \right)\\ = \frac{{{x^3} + 2{x^2}}}{{3{x^2} - 1}}\end{array}\)

Domain of \(f\left( x \right)\) is

\(A = \left( { - \infty ,\infty } \right)\)

Domain of \(g\left( x \right)\) is

\(B = \left( { - \infty ,\infty } \right)\)

But, \(g\left( { \pm \sqrt {\frac{1}{3}} } \right) = 0\)

Thus, domain of \(\left( {f/g} \right)\left( x \right)\) is

\(A \cap B = \left( { - \infty ,\infty } \right);\;\;\;x \ne \pm \sqrt {\frac{1}{3}} \)

The combined function is \(\frac{{{x^3} + 2{x^2}}}{{3{x^2} - 1}}\) and the domain is\(\left( { - \infty ,\infty } \right);\;\;\;x \ne \pm \sqrt {\frac{1}{3}} \).