The Functions S S S And T T T Are Defined As Follows:${ S(x) = 2x + 2 }$ { T(x) = -x^2 + 1 \} Find The Value Of S ( T ( 4 ) S(t(4) S ( T ( 4 ) ]. S ( T ( 4 ) ) = S(t(4)) = S ( T ( 4 )) =

Mar 12, 2025 by ADMIN 189 views

**The Functions s and t: A Mathematical Exploration**

Introduction

In mathematics, functions are used to describe relationships between variables. Two functions, s(x) and t(x), are defined as follows:

${ s(x) = 2x + 2 }$

${ t(x) = -x^2 + 1 }$

In this article, we will explore the composition of these two functions, specifically the value of s(t(4)).

Understanding the Functions s and t

Before we dive into the composition of the functions, let's take a closer look at each function individually.

The Function s(x)

The function s(x) is defined as:

${ s(x) = 2x + 2 }$

This function takes an input value x, multiplies it by 2, and then adds 2 to the result. For example, if we input x = 3, the function s(x) would return:

${ s(3) = 2(3) + 2 = 8 }$

The Function t(x)

The function t(x) is defined as:

${ t(x) = -x^2 + 1 }$

This function takes an input value x, squares it, and then multiplies the result by -1. Finally, it adds 1 to the result. For example, if we input x = 4, the function t(x) would return:

${ t(4) = -(4)^2 + 1 = -15 }$

Finding the Value of s(t(4))

Now that we have a good understanding of the functions s(x) and t(x), let's find the value of s(t(4)).

To do this, we need to input the value of t(4) into the function s(x). We already know that t(4) = -15, so we can plug this value into the function s(x):

${ s(t(4)) = s(-15) }$

Using the definition of the function s(x), we can substitute -15 for x:

${ s(-15) = 2(-15) + 2 }$

Now, we can simplify the expression:

${ s(-15) = -30 + 2 }$

${ s(-15) = -28 }$

Therefore, the value of s(t(4)) is -28.

Conclusion

In this article, we explored the functions s(x) and t(x) and found the value of s(t(4)). We learned how to compose the functions and simplify the resulting expression. This type of problem is an example of a mathematical concept called function composition, which is an important tool in mathematics and computer science.

Real-World Applications

Function composition has many real-world applications, including:

Computer Science: Function composition is used in programming languages to create complex functions from simpler ones.
Engineering: Function composition is used in engineering to model complex systems and relationships between variables.
Economics: Function composition is used in economics to model economic systems and relationships between variables.

Final Thoughts

In conclusion, the functions s(x) and t(x) are two simple functions that can be composed to create a more complex function. By understanding the composition of these functions, we can gain a deeper understanding of mathematical concepts and their real-world applications.

References

[1] "Functions" by Khan Academy
[2] "Function Composition" by Wolfram MathWorld
[3] "Mathematical Functions" by Encyclopedia Britannica

Additional Resources

[1] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon
The Functions s and t: A Mathematical Exploration - Q&A =====================================================

Introduction

In our previous article, we explored the functions s(x) and t(x) and found the value of s(t(4)). We learned how to compose the functions and simplify the resulting expression. In this article, we will answer some common questions related to the functions s(x) and t(x).

Q&A

Q: What is the definition of the function s(x)?

A: The function s(x) is defined as:

${ s(x) = 2x + 2 }$

This function takes an input value x, multiplies it by 2, and then adds 2 to the result.

Q: What is the definition of the function t(x)?

A: The function t(x) is defined as:

${ t(x) = -x^2 + 1 }$

This function takes an input value x, squares it, and then multiplies the result by -1. Finally, it adds 1 to the result.

Q: How do you find the value of s(t(4))?

A: To find the value of s(t(4)), we need to input the value of t(4) into the function s(x). We already know that t(4) = -15, so we can plug this value into the function s(x):

${ s(t(4)) = s(-15) }$

Using the definition of the function s(x), we can substitute -15 for x:

${ s(-15) = 2(-15) + 2 }$

Now, we can simplify the expression:

${ s(-15) = -30 + 2 }$

${ s(-15) = -28 }$

Therefore, the value of s(t(4)) is -28.

Q: What is the difference between the functions s(x) and t(x)?

A: The main difference between the functions s(x) and t(x) is the way they operate on the input value x. The function s(x) multiplies the input value x by 2 and then adds 2 to the result, while the function t(x) squares the input value x, multiplies the result by -1, and then adds 1 to the result.

Q: Can you give an example of how to use the functions s(x) and t(x) in real-world applications?

A: Yes, the functions s(x) and t(x) can be used in real-world applications such as:

Computer Science: Function composition is used in programming languages to create complex functions from simpler ones.
Engineering: Function composition is used in engineering to model complex systems and relationships between variables.
Economics: Function composition is used in economics to model economic systems and relationships between variables.

Q: How do you simplify the expression s(t(4))?

A: To simplify the expression s(t(4)), we need to substitute the value of t(4) into the function s(x). We already know that t(4) = -15, so we can plug this value into the function s(x):

${ s(t(4)) = s(-15) }$

Using the definition of the function s(x), we can substitute -15 for x:

${ s(-15) = 2(-15) + 2 }$

Now, we can simplify the expression:

${ s(-15) = -30 + 2 }$

${ s(-15) = -28 }$

Therefore, the value of s(t(4)) is -28.

Conclusion

In this article, we answered some common questions related to the functions s(x) and t(x). We learned how to compose the functions and simplify the resulting expression. We also discussed the real-world applications of the functions s(x) and t(x).