Question

Find ﻿f+gf+gf+g﻿, ﻿f−gf-gf−g﻿, ﻿fgfgfg﻿, and ﻿fg\frac{f}{g}gf​﻿. Determine the domain for each function.﻿f(x)=5−x2f\left(x\right)=$$5-x^2f(x)=5$$−x2﻿, ﻿g(x)=x2+4x−12g\left(x\right)=$$x^2+4x-12g(x)=x2+4x$$−12

Accepted Answer

Step 1: To find $$ (f+g)(x) $$, add the functions $$ f(x) $$ and $$ g(x) . $$This means you will add $$ 5 - x^2 $$ and $$ x^2 + 4x - 12 . $$Combine like terms to simplify. Step 2: To find $$ (f-g)(x) $$, subtract $$ g(x) $$ from $$ f(x) . $$This involves subtracting $$ x^2 + 4x - 12 $$ from $$ 5 - x^2 . $$Again, combine like terms to simplify. Step 3: To find $$ (fg)(x) $$, multiply the functions $$ f(x) $$ and $$ g(x) . $$This requires distributing $$ 5 - x^2 $$ across $$ x^2 + 4x - 12 $$ and combining like terms. Step 4: To find $$ \left(\frac{f}{g}\right)(x) $$, divide $$ f(x)  by$$ $$ g(x) . $$This means writing $$ \frac{5 - x^2}{x^2 + 4x - 12} . $$Simplify if possible, and identify any restrictions on the domain where the denominator is zero. Step 5: Determine the domain for each function. For $$ f+g $$ and $$ f-g $$, the domain is all real numbers. For $$ fg $$, the domain is also all real numbers. For $$ \frac{f}{g} $$, exclude values that make the denominator zero by solving $$ x^2 + 4x - 12 = 0  to $$find the restricted values.

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.
$f\(\left\)(x\(\right\))=5-x^2$ , $g\(\left\)(x\(\right\))=x^2+4x-12$

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Key Concepts

Function Operations

Domain of a Function

Quadratic Functions

Find f+gf+gf+g, f−gf-gf−g, fgfgfg, and fg\(\frac{f}{g}\)gf. Determine the domain for each function.f(x)=5−x2f\(\left\)(x\(\right\))=5-x^2f(x)=5−x2, g(x)=x2+4x−12g\(\left\)(x\(\right\))=x^2+4x-12g(x)=x2+4x−12