Question

Find

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

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

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

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

and state their domains.

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

Short answer

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

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

(c) The product of the two functions is \(\left( {{\bf{fg}}} \right)\left( {\bf{x}} \right){\bf{ = }}\sqrt {{\bf{3}}{{\bf{x}}^{\bf{2}}}{\bf{ - 3 - }}{{\bf{x}}^{\bf{3}}}{\bf{ + x}}} \)with domain\(\left( {{\bf{ - }}\infty {\bf{, - 1}}} \right){\bf{U}}\left( {{\bf{1,3}}} \right)\).

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

Step-by-step solution

Given data

The provided functions are

\(f\left( x \right) = \sqrt {3 - x} \)

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

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),

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

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

\(A = \left( { - \infty ,3} \right)\)

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

\(B = \left( { - \infty , - 1} \right) \cup \left( {1,\infty } \right)\)

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

\(A \cap B = \left( { - \infty , - 1} \right) \cup \left( {1,3} \right)\)

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

Difference of two functions

From equation (2),

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

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

\(A = \left( { - \infty ,3} \right)\)

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

\(B = \left( { - \infty , - 1} \right) \cup \left( {1,\infty } \right)\)

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

\(A \cap B = \left( { - \infty , - 1} \right) \cup \left( {1,3} \right)\)

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

Product of two functions

From equation (3),

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

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

\(A = \left( { - \infty ,3} \right)\)

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

\(B = \left( { - \infty , - 1} \right) \cup \left( {1,\infty } \right)\)

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

\(A \cap B = \left( { - \infty , - 1} \right) \cup \left( {1,3} \right)\)

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

Division of two functions

From equation (4),

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

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

\(A = \left( { - \infty ,3} \right)\)

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

\(B = \left( { - \infty , - 1} \right) \cup \left( {1,\infty } \right)\)

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

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

\(A \cap B = \left( { - \infty , - 1} \right) \cup \left( {1,3} \right)\)

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