Find f + g f+g f + g , f − g f-g f − g , f g fg f g , and f g \frac{f}{g} g f . Determine the domain for each function. f ( x ) = x f\left(x\right)=\sqrt{x} f ( x ) = x <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> , g ( x ) = x − 5 g\left(x\right)=x-5 g ( x ) = x − 5

Hello. In this video, we are going to be using the two given functions to find the composite function F multiplied by G. And we will identify its domain. The functions given to us are F of X is equal to the square root of X and G of X is equal to X plus 17. How can we use these two functions to find the composite function F multiplied by G? Well, F multiplied by G is telling us that we want to take the F of X function and multiply the G of X function to it. So we will go ahead and take F of X which is the square root of X and multiply G of X which is X plus 17. Currently, this is going to be the simplest form of F multiplied by G. So F multiplied by G is equal to the square root of X multiplied by X plus 17. Now, how about the domain of F multiplied by G? Well, the problem here is that we have a square root function in F multiplied by G. The only values of X that are not allowed to be within a square root are negative numbers because the square root of a negative number produces an imaginary number. So we only want values of X that are going to be greater than, or equal to zero. And that means that the domain is going to be written as bracket zero to positive infinity. So we identified that F multiplied by G is equal to the square root of X multiplied by X plus 17. And the domain are going to be all the numbers from zero to positive infinity, including the number zero. With that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

3. Functions

Function Operations

College Algebra

3. Functions

Function Operations

2:05 minutes

Problem 40

Textbook Question

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

Verified step by step guidance

To find $ (f+g)(x) $, add the functions: $ f(x) + g(x) = \sqrt{x} + (x - 5) $. Simplify the expression to get $ \sqrt{x} + x - 5 $. The domain of $ f(x) = \sqrt{x} $ is $ x \geq 0 $, and the domain of $ g(x) = x - 5 $ is all real numbers. Therefore, the domain of $ f+g $ is $ x \geq 0 $.

To find $ (f-g)(x) $, subtract the functions: $ f(x) - g(x) = \sqrt{x} - (x - 5) $. Simplify the expression to get $ \sqrt{x} - x + 5 $. The domain is the same as $ f+g $, which is $ x \geq 0 $.

To find $ (fg)(x) $, multiply the functions: $ f(x) \cdot g(x) = \sqrt{x} \cdot (x - 5) $. This simplifies to $ x\sqrt{x} - 5\sqrt{x} $. The domain is $ x \geq 0 $ because $ \sqrt{x} $ is only defined for non-negative $ x $.

To find $ \left(\frac{f}{g}\right)(x) $, divide the functions: $ \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{x - 5} $. The domain is $ x \geq 0 $ and $ x \neq 5 $ because the denominator cannot be zero.

Summarize the domains: $ f+g $ and $ f-g $ have domain $ x \geq 0 $, $ fg $ has domain $ x \geq 0 $, and $ \frac{f}{g} $ has domain $ x \geq 0 $ and $ x \neq 5 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. Understanding the domain is crucial because it determines the values that can be used in the function without resulting in undefined expressions, such as division by zero or taking the square root of a negative number.

Square Root Function

The square root function, denoted as f(x) = √x, is defined only for non-negative values of x. This means that the input must be greater than or equal to zero (x ≥ 0) to yield a real number output. Recognizing this restriction is essential when determining the domain of functions involving square roots.

Linear Function

A linear function, such as g(x) = x - 5, is defined for all real numbers. This means there are no restrictions on the input values, and the domain is all real numbers (−∞, ∞). Understanding the nature of linear functions helps in identifying their domains and how they interact with other functions in operations like addition or subtraction.

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