Evaluating Functions - Basic Introduction | Algebra

Summary

This educational video script offers a comprehensive guide to evaluating functions and solving equations involving them. It begins with a simple example of evaluating a function f(x) = x^2 + 5x - 7 at x = 3, demonstrating the process of substitution and calculation. The script then progresses to more complex functions involving multiple variables, like f(x, y) = 2x^2 - 3xy + 4y^2, and illustrates how to find function values given certain inputs. It also tackles the challenge of finding variable values when the function's output is known, using algebraic manipulation and factoring. The video concludes with a problem-solving approach to quadratic equations, either through factoring or utilizing the quadratic formula, providing viewers with a robust understanding of function evaluation and equation solving.

Takeaways

• 📘 To evaluate a function, substitute the given value for the variable in the function's expression.
• 🔢 For functions like f(x) = x^2 + 5x - 7, replace x with the specified value and perform the arithmetic to find f(x).
• 📐 When dealing with functions of two variables, such as f(x, y) = 2x^2 - 3xy + 4y^2, substitute both variables with their given values to evaluate the function.
• 🔍 To find the value of a variable when the function's output is known, set the function equal to the output and solve the resulting equation.
• 📉 For quadratic equations, factoring can be used to find the variable's value, especially when the equation is in the form ax^2 + bx + c = d.
• 🔑 Factoring involves finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient.
• ✅ Verification of solutions is crucial; substitute the found values back into the original function to ensure they yield the expected output.
• 🔄 When factoring is challenging, the quadratic formula y = (-b ± √(b^2 - 4ac)) / (2a) can be used as an alternative method to find the variable's value.
• 📝 In functions with multiple variables, identify which variable's value is being sought and isolate it by substituting known values and solving the resulting equation.
• 🧮 Algebraic manipulation, including expanding, factoring, and simplifying expressions, is key to solving for variables in functions.

Q & A

What is the value of f(x) = x^2 + 5x - 7 when x is 3?

To find the value of f(x) when x is 3, substitute x with 3 in the function: f(3) = 3^2 + 5*3 - 7. This evaluates to 9 + 15 - 7, which equals 17.

How do you evaluate the function f(x, y) = 2x^2 - 3xy + 4y^2 for x = 2 and y = 3?

Substitute x with 2 and y with 3 in the function: f(2, 3) = 2*(2^2) - 3*(2*3) + 4*(3^2). This simplifies to 2*4 - 18 + 4*9, which equals 8 - 18 + 36, resulting in 26.

If f(x) = x^2 - 4x + 9 and f(x) is 5, what is the value of x?

Set f(x) equal to 5: 5 = x^2 - 4x + 9. Rearrange to form a quadratic equation: x^2 - 4x + 4 = 0. Factor the quadratic to get (x - 2)^2 = 0, which gives x = 2.

What is the process to find the value of y in the function f(x, y) = 3x^2 - 4xy + 5y^2 when f(4, y) = 100?

Replace x with 4 and set f(4, y) to 100: 100 = 3*(4^2) - 4*(4*y) + 5*y^2. Simplify and rearrange to form a quadratic in y: 5y^2 - 16y - 52 = 0. Factor or use the quadratic formula to find y = -2 or y = 26/5.

How do you factor a trinomial with a leading coefficient other than one, as seen in the function f(x, y) = 3x^2 - 4xy + 5y^2?

For the trinomial 5y^2 - 16y - 52, find two numbers that multiply to 260 (5*52) and add to -16. These numbers are -26 and 10. Rewrite the middle term and factor by grouping to get (y + 2)(5y - 26).

What is the alternative method to factoring for solving a quadratic equation like 5y^2 - 16y - 52 = 0?

Use the quadratic formula y = [-b ± sqrt(b^2 - 4ac)] / (2a). For the equation 5y^2 - 16y - 52 = 0, substitute a = 5, b = -16, and c = -52 to find the values of y.

How does the quadratic formula help in solving for y in the equation 5y^2 - 16y - 52 = 0?

Apply the quadratic formula: y = [16 ± sqrt((-16)^2 - 4*5*(-52))] / (2*5). This results in y = [16 ± sqrt(256 + 2080)] / 10, which simplifies to y = [16 ± sqrt(2336)] / 10, giving y = 26/5 or y = -2.

What is the significance of setting each factor in a factored equation to zero when solving for variables?

Setting each factor to zero allows you to solve for the variable values that satisfy the equation. In the factored form (y + 2)(5y - 26) = 0, setting y + 2 = 0 and 5y - 26 = 0 gives the solutions y = -2 and y = 26/5.

Why is it important to check your work after solving an equation, as demonstrated in the script?

Checking your work by substituting the found values back into the original equation ensures the solution is correct. For x = 2 in x^2 - 4x + 9 = 5, substituting shows 2^2 - 4*2 + 9 = 5, confirming the solution is accurate.

How does understanding the process of evaluating functions and solving equations help in grasping algebraic concepts?

Evaluating functions and solving equations provide practical applications of algebraic concepts like substitution, factoring, and using the quadratic formula, which are essential for a strong foundation in algebra.

Outlines

00:00

📘 Function Evaluation Basics

This paragraph introduces the concept of function evaluation. It uses the example of a function f(x) = x^2 + 5x - 7 and demonstrates how to find the value of the function when x is 3. The process involves substituting x with 3 in the function, performing the arithmetic, and concluding that f(3) equals 17. The paragraph also presents another function involving two variables, f(x, y) = 2x^2 - 3xy + 4y^2, and shows how to evaluate it when x is 2 and y is 3, resulting in f(2, 3) = 26. The explanation emphasizes the substitution of variable values into the function to find the output.

05:01

🔍 Solving for Variable in a Function

The second paragraph focuses on solving for a variable within a function. It presents a function f(x) = x^2 - 4x + 9 and asks for the value of x when f(x) equals 5. The process involves setting f(x) to 5 and solving the resulting quadratic equation. The paragraph explains how to factor the quadratic equation by finding two numbers that multiply to the constant term and add up to the coefficient of the linear term. The solution is found to be x = 2, which is verified by substituting back into the original function to confirm that f(2) equals 5.

10:01

🧮 Advanced Function Evaluation and Factoring

The third paragraph delves into more complex function evaluation with a function f(x, y) = 3x^2 - 4xy + 5y^2. Given that f(4, y) = 100, the paragraph outlines the steps to solve for y. It involves substituting x with 4 and setting up a new equation in terms of y. The resulting quadratic equation in y is factored by grouping, leading to two possible solutions for y. The paragraph also discusses the use of the quadratic formula as an alternative method for solving such equations, providing the formula and applying it to find the two potential values for y. The correct answer is identified based on the context of the problem.

Keywords

💡Function

In the context of the video, a 'function' is a mathematical concept that relates each element from one set to one element of another set. The video focuses on evaluating functions, which means finding the output of the function for a given input. For instance, when the function is defined as 'f(x) = x^2 + 5x - 7', the process of finding 'f(3)' involves substituting 'x' with '3' and performing the calculations to get the output.

💡Evaluate

To 'evaluate' in mathematics means to calculate the value of an expression or function. The video provides examples of evaluating functions by substituting values for the variables within the function's formula. For example, evaluating 'f(3)' for the function 'f(x) = x^2 + 5x - 7' involves substituting '3' for 'x' and then performing the arithmetic operations to find the value of the function at that point.

💡Variable

A 'variable' in mathematics is a symbol that represents a value that can change. In the video, variables like 'x' and 'y' are used in function definitions, and their values are substituted to evaluate the function. For example, in the function 'f(x, y) = 2x^2 - 3xy + 4y^2', 'x' and 'y' are variables that can be assigned specific values to determine the function's output.

💡Algebra

Algebra is a branch of mathematics that uses symbols and rules to solve equations. The video discusses using algebraic methods to solve for variables within functions. For example, when given 'f(x) = 5' and 'f(x) = x^2 - 4x + 9', the video demonstrates how to use algebra to solve for 'x' by setting up and solving a quadratic equation.

💡Quadratic Equation

A 'quadratic equation' is a polynomial equation of the second degree, typically in the form 'ax^2 + bx + c = 0'. The video explains how to solve such equations, either by factoring or using the quadratic formula, to find the values of variables within functions. An example from the script is solving 'x^2 - 4x + 9 = 5' by rearranging and factoring to find 'x'.

💡Factoring

To 'factor' in mathematics is to express a polynomial as the product of its factors. The video shows how to factor a quadratic equation to find its roots. For instance, when solving 'x^2 - 4x + 9 = 5', the video demonstrates factoring the quadratic expression to '(x - 2)^2' to find the value of 'x'.

💡Quadratic Formula

The 'quadratic formula' is a method for finding the solutions of a quadratic equation. It is given by 'x = [-b ± sqrt(b^2 - 4ac)] / (2a)'. The video provides an example of using the quadratic formula to solve for 'y' in the equation '5y^2 - 16y + 52 = 0', showcasing an alternative method to factoring.

💡Substitute

In the context of the video, 'substitute' refers to the process of replacing variables in a function with specific values to evaluate the function. This is a key step in finding the output of functions for given inputs. For example, substituting 'x' with '3' in 'f(x) = x^2 + 5x - 7' to find 'f(3)'.

💡Trinomial

A 'trinomial' is a polynomial with three terms. The video discusses how to factor trinomials, particularly when the leading coefficient is not one. An example is factoring '5y^2 - 16y + 52' by finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient.

💡Greatest Common Factor (GCF)

The 'GCF' is the largest number that divides two or more numbers without leaving a remainder. In the video, the GCF is used in the process of factoring by grouping. For instance, when factoring '5y^2 + 10y - 26y - 52', the GCF of the first two terms is '5y', and for the last two terms, it is '-26', which helps in factoring the expression.

Highlights

Introduction to evaluating functions by substituting values.

Example of evaluating a function f(x) = x^2 + 5x - 7 at x = 3.

Calculation of f(3) resulting in the value 17.

Explanation of how to evaluate functions with multiple variables, f(x, y) = 2x^2 - 3xy + 4y^2.

Evaluation of f(2, 3) leading to the value 26.

Method for finding the value of x when f(x) is given, using the function f(x) = x^2 - 4x + 9.

Solving the equation 5 = x^2 - 4x + 9 to find x = 2.

Verification of the solution x = 2 by substituting back into the original function.

Introduction to solving for y in a function with two variables, f(x, y) = 3x^2 - 4xy + 5y^2.

Setting up the equation 100 = 3*4^2 - 4*4*y + 5*y^2 to find y.

Transforming the equation into a standard quadratic form to solve for y.

Factoring the quadratic equation to find possible values of y.

Explanation of how to use the quadratic formula as an alternative to factoring.

Application of the quadratic formula to find the values of y.

Identification of the correct answer choice based on the calculated values of y.

Emphasis on the importance of checking work to ensure accuracy in function evaluation.

Transcripts

in this video we're going to focus on

functions evaluating

functions so let's start with this

example if f ofx is x^2 + 5x - 7 then

what is the value of f of

3 so we can see that X is equal to 3 x

is basically the number inside of the

function and we're looking for the value

of the function so what is the value of

f when X is string so all we got to do

is replace x with 3 so instead of

writing X2 we're going to write 3^ 2

instead of writing 5 * X we're going to

write 5 *

3 and we just got to do the math 3 S

that's 3 * 3 which is 9 5 * 3 is

15 and 15 - 7 is 8 and 9 + 8 is 17 so

when X is three F has a value of 17 so D

is the right answer for this

example here's another one if f of x

comma Y is 2x^2 - 3x y + 4

y^2 what is the value of F2 comma

3 so we need to realize that X is

2 and Y is three because they

match so when X is 2 and when Y is 3

what is is the value of the

function so all we got to do is simply

plug in the values so let's go ahead and

evaluate F2 comma 3 so everywhere we see

an X let's replace it with

two and wherever we see a y replace it

with

three so using this

equation this is what we now

have 2^ 2 that's 2 * 2 that's 4 3 * 2 is

6 and 3^ 2 is 9 2 * 4 is 8 6 * 3 is 18 4

* 9 is

36 8 - 18 that's -10 and -10 + 36 is 26

so therefore e is the right

answer here's another problem if f ofx

is = to x^2 - 4x + 9 what is the value

of x when f ofx is

five

so we have the function f f is five and

our goal is to find the value of x how

can we do so so what we need to do is

replace f ofx with

5 so 5 is equal to x^2 - 4x + 9 so at

this point we just have to do some

algebra in order to find the value of x

X so how can we go ahead and uh how can

we solve

this the first thing we should do is

subtract both sides by five we want the

left side to be equal to zero this is a

quadratic equation and we could find the

answer by factoring the trinomial that

we have on the

right so what two numbers multiply to

four but add to the middle coefficient

ne4 the two numbers are going to be -2

and -2 so to factor it it's x - 2 * x -

2 which you can write as x - 2^

2 so to solve this equation you need to

set each factor equal to zero but

because they're the same we just got to

do it once so therefore we can see that

X is equal to

2 now let's go ahead and check the work

that we have

so if we plug in two will it give us

five so it's going to be 2^ 2 - 4 * 2 +

9 2^ 2 is 4 4 * 2 is 8 and 4 - 8 is -4

-4 + 9 is five so clearly we can see

that X is indeed equal to two which

means B is the right

answer here's another example if F ofx

comma Y is 3x^2 - 4x y + 5 y^ 2 and F of

4 comma Y is 100 what is the value of y

so let's make a list of what we have and

what we need to

find so we can see that the first letter

X matches with four so therefore X is

equal to 4

F the entire function is 100 so f is 100

and our goal is to find the value of

y so what we're going to do is we're

going to replace f with 100 replace x

with 4 and find the value of y so let's

go ahead and begin so 100 is equal to

3x^2 but X is 4 so that's 3 * 4^ 2 - 4xy

or 4 * 4 * * y + 5

y^2 so now let's do some Algebra 4^ 2 is

16 4 * 4 is equal to

that 3 * 16 that's

48 and what we have is another quadratic

function so let's subtract both sides by

100

48 minus 100 is - 52 so this is what we

now

have let's get rid of a few

things now what I'm going to do is

rewrite it I'm going to put it in

standard form so this is 5 y^2 - 16 Yus

52 let's see if we can Factor this

expression

so how can we factor a trinomial where

the leading coefficient is not one in

this case it's five what we need to do

is multiply the first number by the last

number so what is 5 *

52 5 * 50 is 250 and 5 * 2 is 10 250 +

10 that's

260 so what two numbers multiply to 260

but add up to -16

260 is basically 26 and

10 now let's put the negative sign here

because we had it here one of these

numbers have to be negative it's either

the 26 or the 10 26 + -10 adds up to POS

16 but - 26 + 10 that's -6 so these are

the two numbers that we need so to

factor this particular trinomial we need

to replace the middle term -16 y

with -26 Y and 10 y now I'm going to

write the 10 y first because five goes

into

10 and 26 goes into

52 now once you replace the middle term

with these two terms your next step is

to factor by

grouping so what we need to do is take

out the GCF or the greatest common

factor in the first two terms and that's

going to be 5

y 5 y^2 / 5 y That's equal to y 10 y / 5

Y is

positive2 now in the last two terms we

need to take out the GCF which is

-26 -26 y / -26 is simply y- 52 /

-26 is

pos2 so notice that we have a common

factor which is y +

2 so if we factor out y + 2 from this

term we're going to get 5

Y and if we take out y + 2 from that

term we're going to get

-26 so now that we have it in factored

form we need to set both factors equal

to zero

so let's set y + 2 = 0 and 5 y - 26 =

0 if we subtract both sides by two we

can see that Y is equal to -2 in this

example or in this equation we got to

add 26 to both sides and then divide by

five so the other answer is 26 over

5 so these are the two possible values

for y

however only one of the answers is

listed so answer Choice B is the right

answer now if you ever get stuck in

terms of factoring let's say if you

don't know what to do when you need to

factor the expression you can always

fall back to the quadratic

formula so let's go ahead and do that

for this

example so a is 5 b is -16 C is 52

X is equal to well in this case it's

going to be y instead of X so Y is equal

to B plus orus s < TK of B ^2 - 4 a c /

2

a so B in this example is

-16 -16 2 is POS 256 that's -6 * -16 a

is 5 and C is

52 2 2 * a a is 5 so 2 * 5 is 10 so what

we have here is postive

16 plus or minus Square t now 4 * 5 is

20 and 20 *

52 that should be like

1040 so this is

256+

1040 over 10

256 + 1040 is

1296 and the square root of

1296 is

36 so this gives us two possible answers

16 + 36 ID 10 and 16us 36 ID 10 16 + 36

is

52 and 52 / 10 if you divide both

numbers by two you're going to get 26

over

5 16 - 36 is -20 -20 ID 10 is -2 and

that's the answer that we have so

whenever you get a trinomial you can

either Factor it or you can use the

quadratic formula to find the value of x

or y