Evaluating composite functions | Mathematics III | High School Math | Khan Academy

Summary

The video script explains the concept of function composition using two functions, g(x) = x^2 + 5x - 3 and h(y) = 3(y - 1)^2 - 5. It demonstrates how to find h(g(-6)) by first calculating g(-6), which results in 3, and then using this output as the input for h, yielding h(3) = 7. The script emphasizes the importance of understanding function composition notation and provides a step-by-step approach to solving such problems.

Takeaways

• 📘 The function g(x) is defined as x squared plus five x minus three.
• 📙 The function h(y) is defined as three times (y minus one) squared minus five.
• 🔁 Function composition is represented by a circle symbol between two functions, indicating that one function is applied after the other.
• 🤔 The process of function composition involves evaluating the inner function first and then using its result as the input for the outer function.
• 🔢 To find h(g(-6)), first calculate g(-6) by substituting -6 into the function g(x).
• 🧮 After evaluating g(-6), the result is 3, which is then used as the input for the function h(y).
• 📐 The calculation for h(3) involves squaring the result from g(-6), multiplying by three, and then subtracting five.
• 📈 The final result of h(g(-6)) is 7, which is obtained by following the steps of function composition.
• 📝 Understanding function composition is crucial for solving problems that involve nested functions.
• 📖 The script emphasizes the importance of taking a step-by-step approach to solve complex problems involving function composition.

Q & A

What is the mathematical expression for g(x) as described in the transcript?

The mathematical expression for g(x) is g(x) = x^2 + 5x - 3.

What is the mathematical expression for h(y) as described in the transcript?

The mathematical expression for h(y) is h(y) = 3(y - 1)^2 - 5.

What does the function composition symbol '∘' represent?

The function composition symbol '∘' represents the application of one function to the result of another function.

How is h(g(-6)) expressed in terms of function composition?

h(g(-6)) can be expressed as h(g(x)) where x is -6, which means you first apply g(x) to -6 and then apply h(y) to the result.

What is the first step in calculating h(g(-6))?

The first step in calculating h(g(-6)) is to find the value of g(-6) by substituting -6 into the function g(x).

What is the value of g(-6) after substituting -6 into the function g(x)?

The value of g(-6) is calculated as (-6)^2 + 5*(-6) - 3, which equals 36 - 30 - 3, resulting in 3.

After finding g(-6), what is the next step in calculating h(g(-6))?

The next step is to substitute the value of g(-6), which is 3, into the function h(y) to find h(3).

What is the value of h(3) after substituting 3 into the function h(y)?

The value of h(3) is calculated as 3*(3 - 1)^2 - 5, which equals 3*(2)^2 - 5, resulting in 12 - 5, which is 7.

What is the final result of h(g(-6))?

The final result of h(g(-6)) is 7, after substituting -6 into g(x) and then substituting the result into h(y).

Why is it important to understand function composition?

Understanding function composition is important because it allows you to analyze and solve problems involving multiple functions and their interactions.

What advice does the voiceover give for dealing with function composition?

The voiceover advises to take a breath and take it one step at a time when dealing with function composition to avoid confusion.

Outlines

00:00

📘 Understanding Function Composition

The paragraph introduces the concept of function composition using the functions g(x) = x^2 + 5x - 3 and h(y) = 3(y - 1)^2 - 5. It explains that h(g(-6)) represents the composition of h and g, where the output of g(-6) is used as the input for h. The paragraph emphasizes the importance of understanding the composition symbol, which is a circle between the two functions. It suggests rewriting the composition in a more intuitive way, such as h(g(-6)) as h(g(-6)) = h(g(-6)). The process involves calculating g(-6) first, which is -6^2 + 5*(-6) - 3, resulting in 3. Then, this result is used as the input for h, leading to h(3), which is calculated as 3*(3 - 1)^2 - 5, simplifying to 7. The final result, h(g(-6)), is therefore 7, demonstrating the step-by-step approach to function composition.

Keywords

💡Function Composition

Function composition is a process in mathematics where one function is applied to the result of another function. In the video, function composition is the central theme, demonstrated by the calculation of 'h of g of negative six.' The script explains that this means to first compute 'g of negative six' and then use that result as the input for the function 'h.' The concept is crucial for understanding how multiple functions can be combined to create a new function.

💡g of x

In the video, 'g of x' is defined as a mathematical function expressed as g(x) = x^2 + 5x - 3. This function is used to demonstrate the concept of function composition. The script guides the viewer through the process of evaluating 'g of negative six,' which involves substituting -6 for x in the function and performing the arithmetic to find the output.

💡h of y

Similarly, 'h of y' is another function introduced in the video, defined as h(y) = 3(y - 1)^2 - 5. This function is used in conjunction with 'g of x' to illustrate function composition. After finding the result of 'g of negative six,' the script shows how to input this result into 'h of y' to find the final output.

💡Negative Six

Negative six is the specific input value used in the video to demonstrate the process of function composition. The script uses this number to show how it is first input into the function 'g' and then the result is used as an input for the function 'h.' It serves as a practical example to help viewers understand how to apply function composition with real numbers.

💡Output

The term 'output' in the video refers to the result of applying a function to an input value. The script explains that after calculating 'g of negative six,' the output is then used as an input for the function 'h.' The final output of 'h of g of negative six' is the ultimate result of the function composition process, which is found to be seven in this example.

💡Arithmetic

Arithmetic is the mathematical operation of addition, subtraction, multiplication, and division. The video script uses arithmetic extensively to calculate the values of 'g of negative six' and 'h of three.' The arithmetic steps are explained in detail to show how the functions are evaluated, which is essential for understanding function composition.

💡Squaring

Squaring is the operation of multiplying a number by itself, represented as x^2. In the script, squaring is used when calculating 'g of negative six,' where (-6)^2 is computed as part of the function 'g of x.' The concept is fundamental to the algebraic manipulation required in the function composition process.

💡Substitution

Substitution is the process of replacing a variable with a specific value. The video script demonstrates substitution by replacing 'x' with -6 in the function 'g of x' and then using the result to replace 'y' in the function 'h of y.' This is a key step in function composition, as it allows for the chaining of function evaluations.

💡Intuitive

The term 'intuitive' in the video refers to the ease of understanding or the naturalness with which a concept can be grasped. The script contrasts the function composition notation with a more intuitive notation, suggesting that while the circle notation might seem strange at first, it represents the same concept of function composition that can be more easily visualized with a step-by-step approach.

💡Stress

In the context of the video, 'stress' is used to describe the potential anxiety or confusion that learners might feel when encountering function composition for the first time. The script advises viewers not to stress over the unfamiliar composition notation, emphasizing a calm and methodical approach to understanding the concept.

💡One Step at a Time

This phrase is used in the video to encourage a methodical approach to solving problems involving function composition. The script suggests breaking down the process into smaller, more manageable steps, which can help in understanding and evaluating complex mathematical functions. This approach is exemplified by the step-by-step calculation shown in the script.

Highlights

Introduction to function composition symbol (∘).

Explanation of rewriting h(g(-6)) as g(-6) followed by h of that result.

Emphasis on the importance of function composition notation in mathematics.

Guidance on how to approach function composition step by step.

Calculation of g(-6) by substituting -6 into the function g(x).

Step-by-step evaluation of g(-6) resulting in the value 3.

Simplification of h(g(-6)) to h(3) after finding g(-6) equals 3.

Calculation of h(3) by substituting 3 into the function h(y).

Evaluation of h(3) resulting in the final value of 7.

Final conclusion that h(g(-6)) equals 7.

Process of taking the output from g and inputting it into h.

Reassurance for beginners that function composition may seem complex but can be simplified.

Advice to take a breath and tackle function composition one step at a time.

Illustration of how function composition works using a specific example.

Demonstration of the practical application of function composition in a mathematical context.

Clarification that function composition is a fundamental concept in calculus and higher mathematics.

Encouragement for viewers to become familiar with the concept of function composition for future mathematical studies.

Transcripts

- [Voiceover] So, we're told that g of x

is equal to x squared plus five x minus three

and h of y is equal to three times

y minus one squared, minus five.

And then, we're asked,

what is h of g of negative six?

And the way it's written might look a little strange to you.

This little circle that we have in between the h and the g,

that's our function composition symbol.

So, function,

function composition,

composition,

composition symbol.

And one way to rewrite this,

it might make a little bit more sense.

So, this h of g of negative six.

You could rewrite this as,

this is going to be the same thing as

g of negative six,

and then h of that.

So, h of g of negative six.

Notice, I spoke this out the same way that I said this.

This is h of g of negative six.

This is h of g of negative six.

I find the second notation far more intuitive,

but it's good to become familiar with this

function composition notation, this little circle,

because you might see that sometime and

you shouldn't stress, it's just the same thing as

what we have right over here.

Now, what is h of g of negative six?

Well, we just have to remind ourselves that this means

that we're going to take the number negative six,

we're going to input it into our function g,

and then that will output g of negative six,

whatever that number is, we'll figure it out in a second,

and then we're going to input that into our function h.

We're going to input that into our function h.

And then, what we output is going to be

h of g of negative six, which is what we want to figure out.

h of g

of negative six.

So, we just have to do it one step at a time.

A lot of times, when you first start looking at

these function composition, it seems really convoluted

and confusing, but you just have to,

I want you to take a breath and take it one step at a time.

Well, let's figure out what g of negative six is.

It's going to evaluate to a number in this case.

And then, we input that number into the function h,

and then we'll figure out another ...

that's going to map to another number.

So, g of negative six.

Let's figure that out.

g of negative six is equal to

negative six squared,

plus five times negative six, minus three,

which is equal to positive 36, minus 30, minus three.

So, that's equal to what?

36 minus 33, which is equal to three.

So, g of negative six is equal to three.

g of negative six is equal to three.

g of negative six is equal to three.

You input negative six into g, it outputs three.

And so, h of g of negative six has now simplified

to just h of three because g of negative six is three.

So, let's figure out what h of three is.

h of three ... notice, whatever we outputted from g,

we're inputting that now into h.

So, that's the number three, so h of three is going to be

three times three minus one,

three minus one squared, minus five,

which is equal to three times two squared,

this is two right over here, minus five,

which is equal to three times four minus five,

which is equal to 12 minus five,

which is equal to seven.

And we're done.

So, you input negative six into g, you get three.

And then, you take that output from g

and you put it into h and you get seven.

So, this right over here is seven.

All of this has come out to be equal to seven.

So, h of g of negative six is equal to seven.

h of g of negative six is equal to seven.

Input negative six into g, take that output

and input it into h, and you're gonna get seven.