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 ) = 6 x 2 − x − 1 f\left(x\right)=6x^2-x-1 f ( x ) = 6 x 2 − x − 1 , g ( x ) = x − 1 g\left(x\right)=x-1 g ( x ) = x − 1

Hello. In this video, we are going to be using the two given functions to find the composite function F plus G. The functions given to us are F of X is equal to 11 X squared minus three X minus seven. And the second function given to us is G of X is equal to three X minus four. So how can we use these two functions to find the overall function F plus G? Well, F plus G is telling us that we need to take the F of X function and add G of X to it. So in order to find this composite function, we will use F of X which again is 11 X squared minus three X minus seven and add the G of X function which will be three X minus four. What we will need to do now is we will need to simplify. There are a few terms that we can combine together, we can combine negative three X and positive three X. And these terms will cancel each other out. We can also combine negative seven and negative four, negative seven minus four will leave us with negative 11. So we can simplify this function to be 11 X squared minus 11. This is going to be the simplest form of the composite function F plus G. Now what about its domain? Well, F of F plus G is a polynomial function and polynomial functions always have the domain of all real numbers. So that tells us that the domain is going to be all real numbers which will be negative infinity to positive infinity. So F plus G is equal to 11 X squared minus 11 and its domain will be from negative infinity to positive infinity. With that being said, the solution to this problem is going to be C 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

2:08 minutes

Problem 36

Textbook Question

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.
$f\left(x\right)=6x^2-x-1$ , $g\left(x\right)=x-1$

Verified step by step guidance

To find $ (f+g)(x) $, add the functions $ f(x) $ and $ g(x) $: $ (f+g)(x) = f(x) + g(x) = (6x^2 - x - 1) + (x - 1) $. Combine like terms to simplify.

To find $ (f-g)(x) $, subtract $ g(x) $ from $ f(x) $: $ (f-g)(x) = f(x) - g(x) = (6x^2 - x - 1) - (x - 1) $. Distribute the negative sign and combine like terms.

To find $ (fg)(x) $, multiply the functions $ f(x) $ and $ g(x) $: $ (fg)(x) = f(x) \cdot g(x) = (6x^2 - x - 1)(x - 1) $. Use the distributive property (FOIL method) to expand the expression.

To find $ \left(\frac{f}{g}\right)(x) $, divide $ f(x) $ by $ g(x) $: $ \left(\frac{f}{g}\right)(x) = \frac{6x^2 - x - 1}{x - 1} $. Simplify the expression if possible by factoring the numerator and checking for common factors with the denominator.

Determine the domain for each function: For $ f+g $ and $ f-g $, the domain is all real numbers since there are no restrictions. For $ fg $, the domain is also all real numbers. For $ \frac{f}{g} $, the domain excludes $ x = 1 $ because it makes the denominator zero, which is undefined.

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Key Concepts

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

Function Operations

Function operations involve combining two or more functions through addition, subtraction, multiplication, or division. For example, if f(x) and g(x) are two functions, their sum f+g is defined as (f+g)(x) = f(x) + g(x). Understanding these operations is crucial for manipulating and analyzing functions in algebra.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions, like f(x) = 6x^2 - x - 1, the domain is all real numbers. However, for rational functions, such as g(x) = x - 1, the domain must exclude values that make the denominator zero.

Rational Functions

Rational functions are ratios of two polynomials. The domain of a rational function is determined by identifying values that would make the denominator zero, as these values are undefined. For instance, in the function g(x) = x - 1, the domain is all real numbers except where the denominator equals zero, which is critical for ensuring valid function outputs.

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Find f+gf+gf+g, f−gf-gf−g, fgfgfg, and fg\frac{f}{g}gf. Determine the domain for each function.f(x)=6x2−x−1f\left(x\right)=6x^2-x-1f(x)=6x2−x−1, g(x)=x−1g\left(x\right)=x-1g(x)=x−1

Key Concepts

Function Operations

Domain of a Function

Rational Functions

Watch next

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.
$f\left(x\right)=6x^2-x-1$ , $g\left(x\right)=x-1$