Function Operations

Summary

The video script explains the operations on two functions, f(x) = 2x + 5 and g(x) = x^2 - 4. It demonstrates how to find the sum (f + g = x^2 + 2x + 1), difference (f - g = -x^2 + 2x + 9), and product (f * g = 2x^3 + 5x^2 - 8x - 20) of the functions. It then discusses the domain of these functions, highlighting that polynomials have a domain of all real numbers, while fractions require excluding values that make the denominator zero. Examples are given to calculate specific values of f and g for certain x inputs, emphasizing the process of substitution and arithmetic operations.

Takeaways

• 🔢 The function f(x) = 2x + 5 and g(x) = x^2 - 4 are given, and their sum f + g results in x^2 + 2x + 1 after combining like terms.
• ➖ The difference f - g is calculated as -x^2 + 2x + 9 by subtracting g(x) from f(x) and simplifying.
• 🔗 The product f * g is found by multiplying the two functions using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.
• 🌐 The domain of f + g, f - g, and f * g is all real numbers since they are polynomials without restrictions.
• 🚫 The domain of f / g is restricted because it involves a fraction; x cannot be the values that make the denominator zero (x ≠ -2, 2).
• 📐 The domain for functions involving fractions is determined by setting the denominator equal to zero to find the vertical asymptotes.
• 📝 Interval notation is used to express the domain of functions with restrictions, excluding the values that make the denominator zero.
• 🔢 To find f(a) + g(b), substitute 'a' into f(x) and 'b' into g(x), then add the results.
• 🔄 To calculate f(a) * g(b), first find the values of f(a) and g(b) separately, then multiply them together.
• 📘 The script provides a comprehensive guide on how to perform operations on functions and determine their domains.

Q & A

What is the function f(x) as described in the transcript?

The function f(x) is described as f(x) = 2x + 5.

What is the function g(x) as described in the transcript?

The function g(x) is described as g(x) = x^2 - 4.

How do you find the sum of the functions f(x) and g(x)?

You add the two functions together, which results in (2x + 5) + (x^2 - 4), and then combine like terms to get x^2 + 2x + 1.

What is the result of f(x) - g(x) according to the transcript?

The result of f(x) - g(x) is -x^2 + 2x + 9.

How is the product of f(x) and g(x) calculated?

The product f(x) * g(x) is calculated by multiplying (2x + 5) with (x^2 - 4) using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.

What is the domain of the function f(x) + g(x)?

The domain of f(x) + g(x) is all real numbers, as it is a polynomial without restrictions.

What is the domain of the function f(x) - g(x)?

The domain of f(x) - g(x) is also all real numbers, as it is a polynomial without restrictions.

What is the domain of the function f(x) * g(x)?

The domain of f(x) * g(x) is all real numbers, as it is a polynomial without restrictions.

How do you determine the domain of f(x) / g(x)?

The domain of f(x) / g(x) is all real numbers except where the denominator g(x) equals zero, which is when x = -2 or x = 2.

What is the interval notation for the domain of f(x) / g(x)?

The interval notation for the domain of f(x) / g(x) is (-∞, -2) U (-2, 2) U (2, ∞).

What is the value of f(2) + g(3) as described in the transcript?

The value of f(2) + g(3) is calculated as (4*2 + 5) + (8 - 3^2) which equals 13 - 1, resulting in 12.

What is the value of f(-2) * g(2) according to the transcript?

The value of f(-2) * g(2) is calculated as (4*(-2) + 5) * (8 - 2^2) which equals (-3) * 4, resulting in -12.

Outlines

00:00

📘 Operations and Domain of Functions

This paragraph explains the operations on two functions, f(x) = 2x + 5 and g(x) = x^2 - 4. It details how to find the sum (f + g = x^2 + 2x + 1), difference (f - g = -x^2 + 2x + 9), and product (f * g = 2x^3 + 5x^2 - 8x - 20) of the functions. It also discusses the domain of these functions, stating that since they are polynomials without radicals or fractions, their domain is all real numbers (-∞, ∞). However, for the division f/g, where f/g = (2x + 5) / (x^2 - 4), the domain is restricted to all real numbers except x = ±2, as these values make the denominator zero, leading to undefined function values. The domain is expressed in interval notation as (-∞, -2) U (-2, 2) U (2, ∞).

05:03

🔢 Evaluating Functions at Specific Points

The second paragraph demonstrates how to evaluate functions at specific points. It uses the functions f(x) = 4x + 5 and g(x) = 8 - x^2 to calculate f(2) + g(3) and f(-2) * g(2). For f(2) + g(3), it substitutes x with 2 in f(x) and x with 3 in g(x), resulting in 13 - 1 = 12. For f(-2) * g(2), it first calculates f(-2) = -3 and g(2) = 4, then multiplies these values to get -12. The process involves substituting the given x values into the function's formulas and performing the arithmetic operations to find the results.

Keywords

💡Function

In the context of the video, a function refers to a mathematical relation that maps each element from one set to one element of another set. The video discusses functions such as 'f of x' and 'g of x', where 'f of x' is defined as 'two x plus five' and 'g of x' as 'x squared minus four'. Functions are fundamental to the video's theme as they are the basis for the operations performed.

💡Polynomial

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The video mentions polynomials when discussing the domain of 'f plus g', which is a polynomial expression 'x squared plus two x plus one', indicating that its domain is all real numbers.

💡Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The video explains the domain in relation to different functions, stating that for polynomials like 'f plus g', the domain is all real numbers. However, for 'f divided by g', the domain excludes values that make the denominator zero, hence x cannot be -2 or 2.

💡Range

Although not explicitly mentioned in the script, the range of a function is the set of all possible output values (y-values) that result from the input values in the domain. The concept is implicitly discussed when the video talks about the domain of 'f divided by g', where the range would be all real numbers except for undefined values at the vertical asymptotes.

💡Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never reaches, typically occurring where the denominator of a rational function is zero. The video script discusses vertical asymptotes in the context of 'f divided by g', where x cannot be -2 or 2 because these values would make the denominator zero, resulting in vertical asymptotes.

💡Factoring

Factoring is the process of breaking down a polynomial into a product of other polynomials. The video uses factoring when determining the domain of 'f divided by g', where 'x squared minus four' is factored into '(x plus two) times (x minus two)' to find the values of x that are not in the domain.

💡Interval Notation

Interval notation is a method of describing a set of numbers on a number line using parentheses and brackets. The video script uses interval notation to express the domain of 'f divided by g', indicating that the domain is all real numbers except for -2 and 2, written as '(-∞, -2) U (-2, 2) U (2, ∞)'.

💡Combining Like Terms

Combining like terms is a process in algebra where terms that have the same variables raised to the same power are added or subtracted. In the video, this concept is used when adding 'f of x' and 'g of x' to find 'f plus g', where the like terms 'five' and 'negative four' are combined to get 'one'.

💡FOIL Method

The FOIL method is a mnemonic for multiplying two binomials, standing for First, Outer, Inner, Last. The video uses the FOIL method to multiply 'f of x' and 'g of x' to find 'f times g', resulting in the expression '2x cubed plus 5x squared minus 8x minus 20'.

💡Substitution

Substitution in mathematics is the process of replacing a variable with a specific value. The video script uses substitution when calculating 'f of 2' and 'g of 3', where the value of x is replaced with 2 and 3, respectively, to find the values of the functions at those points.

Highlights

Function f(x) is defined as 2x + 5.

Function g(x) is defined as x^2 - 4.

Sum of functions f and g results in x^2 + 2x + 1 after combining like terms.

Difference of functions f and g results in -x^2 + 2x + 9 after distributing and combining constants.

Product of functions f and g is calculated using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.

Domain of f + g, f - g, and f * g is all real numbers since they are polynomials without restrictions.

Domain of f / g is restricted to all real numbers except x = -2 and x = 2, as these make the denominator zero.

Domain of a function with a fraction is determined by setting the denominator equal to zero to find restrictions.

Vertical asymptotes at x = -2 and x = 2 for the function f / g, indicating points of discontinuity.

Domain of f / g is expressed in interval notation as (-∞, -2) U (-2, 2) U (2, ∞).

Function f(x) = 4x + 5 is given as an example for evaluating f(2).

Function g(x) = 8 - x^2 is used to evaluate g(3).

f(2) + g(3) is calculated by substituting x values into their respective functions and summing the results.

f(-2) * g(2) is computed by finding the values of f at x = -2 and g at x = 2, then multiplying these values.

The process of evaluating functions at specific points involves substitution and arithmetic operations.

Transcripts

let's say that f of x

is equal to two x plus five

and g of

x let's say g of x is

x squared minus four

perform the indicated operations

so what is f plus g

what's the sum of the two functions all

you gotta do is add them 2x plus five

plus x squared minus four

and combine like terms

so all we can combine is five and

negative four

which adds up to one so it's x squared

plus two x

plus one

and so that's the sum of f and g

now what about f minus g

f is two x plus five

and then it's gonna be minus

x squared minus four

so this is going to be negative x

squared

plus two x

and then we have five

minus negative four

five minus negative four is like five

plus four

which is nine

so that's equal to f minus g

now what about f times g

this is just going to be two x plus 5

times x squared minus 4.

and we can go ahead and foil it 2x times

x squared

that's 2x cubed

and then 2x times negative 4 that's

negative 8x

and then we have 5 times

x squared that's

5x squared

and then 5 times negative 4 it's

negative 20.

so in standard form it's 2x cubed

plus 5x squared

minus 8x minus 20.

now what is the domain

of the three functions that we found

let's start with f plus g

what is the domain for that

whenever you have a polynomial be it a

binomial trinomial or many terms where

you don't have any fractions no radicals

or

logarithmic functions

the domain will be all real numbers

there's no restrictions on the value of

x in this expression x can be anything

so for these three functions f plus g f

minus g and f times g

we don't have any fractions or radicals

so the domain is all real numbers it's

negative infinity to infinity

now what about f

divided by g

what's the domain for this

f divided by g is simply two x plus five

divided by x squared minus 4.

now that we have a fraction

the domain is restricted

is not all real numbers

to find the domain

you want to find the x values

that do not exist

the values that x cannot be

to do that

set the denominator equal to zero

x squared minus four cannot be zero if

it is we're going to get a vertical

asymptote which means it's undefined at

that point

now we can factor x squared minus four

it's x plus two times x minus two

so therefore x

cannot equal negative two

and it can't equal two these are the

vertical asymptotes

but they're also

infinite discontinuities

there's no we can't plug in negative two

for x the function will be undefined

anytime you have a zero on the bottom

it's undefined

so how can we write the domain using

interval notation if x cannot equal

negative two or two

it's going to be from negative infinity

to negative two

union negative two to two

union

two to infinity

so here's another example let's say

if we have the function one over x minus

three

x minus three the denominator cannot be

zero

so x cannot be three

therefore the domain is everything

except three that's how you write it

let's say if we have this

x

cannot equal 4

and it can't equal

negative 3. just change the sign

if you set x minus four to zero x

you'll get four facts

now to write the domain it's gonna be

negative infinity to negative three

negative three comes before four

union negative three is a four

union 4 to infinity

so that's the domain for this function

so let's say that

f of x

is equal to 4x plus 5.

and g of x

let's say it's equal to 8

minus x squared

what is f of 2

plus g of three

to find f of two

we need to replace x with two in the

equation four x plus five

so that's four times two plus five

and that is a terrible looking two

to find g of 3 we need to plug it into

that equation so it's plus

8 minus

3 squared

and then let's add 4 times 2 is 8

and 3 squared

is nine eight plus five is thirteen

eight minus nine is negative one

thirteen minus one is twelve

so that's the value of f of two plus g

of three

you just find the two values

and you add them

let's try another example

so what is

f of negative 2

multiplied

by

g of two

so let's do it separately let's find f

of negative two first

using this formula

so it's gonna be four times negative two

plus five

which is negative eight plus five

and so that's negative three

now let's calculate g of two

using this equation

so that's eight minus two squared

two squared is four eight minus four is

four

so now that we have these two values

let's replace f of negative two which

is negative three and let's replace g of

two with four

so negative three times four is negative

twelve

so that's the value of f of negative two

times g of negative two i mean times g

of positive two

so all you gotta do is simply

find the values

and then multiply them