Find a function’s output value when given an input value

Summary

This script explains the process of finding an input value for a given output in a linear function. It uses the function f(x) = 2x - 2 and demonstrates how to solve for x when f(x) equals -8. The solution involves setting up an equation, isolating the variable, and finding x = -3, which means an input of -3 yields an output of -8. The script also clarifies the distinction between input (x-value) and output (y-value) and how to represent the solution in function notation and as an ordered pair.

Takeaways

• 🔍 The script explains a method to find the input value of a function given a specific output value.
• 📚 The function in question is f(x) = 2x - 2, and the goal is to find the x value that results in an output of -8.
• 🔄 The first step is to set the function equal to the given output value, replacing f(x) with -8.
• 📝 The equation is then rewritten as -8 = 2x - 2, which is a linear equation.
• 🧩 The next step involves solving the linear equation by adding 2 to both sides, resulting in -6 = 2x.
• ➗ To isolate x, both sides of the equation are divided by 2, leading to x = -3.
• 📌 The solution indicates that an input value of -3 for the function will produce an output value of -8.
• 📈 The input and output values are clarified, with x being the input and Y being the output.
• 📑 The script mentions that answers can be given in function notation, where f(x) = y.
• 📝 In function notation, the solution is expressed as f(-3) = -8, which corresponds to an ordered pair.
• 📍 The ordered pair (-3, -8) represents the coordinates where the function intersects with the line y = -8.
• 📘 The script serves as a tutorial on solving for an input value in a linear function given an output value.

Q & A

What is the given function f(x) in the script?

The given function is f(x) = 2x - 2.

What is the specific output value we are trying to find the input for in the script?

We are trying to find the input value for the output value of -8.

What is the first step in solving for x when f(x) = -8?

The first step is to replace f(x) with -8 in the equation, resulting in -8 = 2x - 2.

How do we rewrite the function notation f(x) to represent the output value?

We replace f(x) with the output value, so it becomes -8 = 2x - 2.

What type of equation are we dealing with in this problem?

We are dealing with a basic linear equation.

What operation is performed to isolate 2x in the equation -8 = 2x - 2?

We add 2 to both sides of the equation to isolate 2x, resulting in -6 = 2x.

How do we solve for x after isolating the term 2x?

We divide both sides of the equation by 2, which gives us x = -3.

What is the input value of x that results in an output of -8 for the function f(x)?

The input value of x that results in an output of -8 is -3.

How is the input and output values related in function notation?

In function notation, the input value is represented by x and the output value is represented by y, so f(x) = y.

What is the ordered pair that corresponds to the function f(x) when x = -3?

The ordered pair is (-3, -8), where -3 is the x-coordinate and -8 is the y-coordinate.

How can the solution be expressed in function notation?

The solution can be expressed in function notation as f(-3) = -8.

Outlines

00:00

🔍 Finding Input for a Given Output in a Linear Function

This paragraph explains the process of finding an input value for a function given a specific output value. The function in question is f(x) = 2x - 2, and the goal is to find the value of x that results in an output of -8. The explanation begins by setting f(x) equal to -8 and then solving the resulting linear equation, -8 = 2x - 2, through basic algebraic steps. The solution involves adding 2 to both sides to isolate the term with x, resulting in -6 = 2x, and then dividing by 2 to find x = -3. The paragraph concludes by emphasizing the relationship between the input value (x) and the output value (y) in function notation, and how to express the solution as an ordered pair (-3, -8) that corresponds to the function.

Keywords

💡Input value

The input value is the variable or number that you put into a function to get a result. In the video's context, the input value is 'x', which is the value that, when substituted into the function, will give a specific output. The script explains how to find the input value 'x' that results in the function 'f(x) = 2x - 2' producing an output of -8.

💡Output value

The output value is the result you get after applying the input value to a function. In the script, the output value is -8, which is the result the speaker is trying to achieve by finding the correct input value for the given function. The video emphasizes the relationship between the input and output values in the context of a function.

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. In the script, 'f(x) = 2x - 2' is the function being discussed, and the process of finding the input value that yields a specific output is the focus of the explanation.

💡Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The script simplifies the function into a linear equation (-8 = 2x - 2) to solve for the input value 'x' that results in the desired output.

💡Solving an equation

Solving an equation involves finding the value(s) of the variable(s) that make the equation true. The script demonstrates the process of solving the linear equation to find the value of 'x' that satisfies the condition of the function producing an output of -8.

💡Algebraic manipulation

Algebraic manipulation refers to the process of changing an equation from one form to another without changing the solutions of the equation. In the script, the speaker uses algebraic manipulation to isolate the variable 'x' by adding and dividing.

💡Basic linear equation

A basic linear equation is a simple form of a linear equation that can be easily solved by applying basic algebraic principles. The script identifies the equation -8 = 2x - 2 as a basic linear equation, which is then solved to find the input value for 'x'.

💡Function notation

Function notation is a way to represent a function with a single letter, often 'f', followed by the variable in parentheses. In the script, the function is represented as 'f(x)', and the speaker explains how to use function notation to express the relationship between the input and output values.

💡Ordered pair

An ordered pair in mathematics is a pair of numbers (x, y) that can represent a point in a coordinate system. The script uses the ordered pair (-3, -8) to represent the input and output values that satisfy the function 'f(x) = 2x - 2'.

💡X-coordinate and Y-coordinate

In a coordinate system, the x-coordinate represents the horizontal position of a point, while the y-coordinate represents the vertical position. The script uses the terms to describe the input value (x-coordinate) and the output value (y-coordinate) in the context of the function.

Highlights

The process of finding an input value for a given output value in a function is explained.

The function f(x) = 2x - 2 is used as an example to demonstrate the process.

The goal is to find the value of x when f(x) equals -8.

The function notation f(x) is replaced with the output value -8 to set up the equation.

The equation -8 = 2x - 2 is formed by substituting the function with the output value.

Solving the linear equation involves basic algebraic steps.

Adding 2 to both sides of the equation simplifies it to -6 = 2x.

Dividing both sides by 2 yields the solution x = -3.

The input value x = -3 generates the output value f(x) = -8 for the given function.

The relationship between input and output values in a function is clarified.

The importance of understanding the function notation f(x) = y is emphasized.

The solution is presented in function notation as f(-3) = -8.

The concept of ordered pairs in the context of functions is introduced.

The ordered pair (-3, -8) corresponds to the input and output values of the function.

The x-coordinate represents the input value, and the y-coordinate represents the output value.

The process is summarized to ensure understanding of input and output values in functions.

Transcripts

what if we want to find an input value

when we're given a specific output value

let's say we have the function f of X

equals 2x minus 2 and we need to find

the value of x such that f of X equals

negative 8 our first step is to replace

f of X with negative 8 now let's rewrite

our function so f of x equals 2x minus 2

now remember the notation f of X that's

equivalent to Y which is equivalent to

our output value so everywhere I see the

notation f of X I'm going to replace it

with the number negative 8

so that means negative 8 is going to be

on the left side of my equation so now I

have the equation negative 8 equals 2x

minus 2 well hey look at that that's

just a basic linear equation good thing

we already know how to solve that so

that's step 2

so my equation is negative 8 equals 2x

minus 2 so to solve this I would add 2

to both sides

that would give me negative 6 equals 2x

then I would divide both sides by 2 and

I'd get x equals negative 3 so this

tells us that my function f of x equals

negative 8 when the value of x is

negative 3 so that means I need an input

value of negative 3 in order to generate

an output value of negative 8 let's make

sure we have those input and output

values straight my input value is the x

value that's negative 3 that's what I

would need to input in the function in

order to generate an output value of

negative 8

remember the input value is our x value

and the output value is our Y value

sometimes you might be asked to give

your answer in function notation

so remember f of x equals y that's our

general function notation so now if I

want to write f of X I'm going to

replace the X with the input value

negative 3 and then that's going to be

equal to my

output value of negative 8 so for

function notation I would say F of

negative 3 equals negative 8 that

corresponds to an ordered pair that

belongs to this function so since

negative 3 is in the spot of the input

value that is my x-coordinate and since

it equals an output value of negative 8

I use negative 8 as my y-coordinate so

my ordered pair is negative 3 negative 8