Evaluating Composite Functions Find S(t(3))

$\begin{array}{l}
s(x)=2 x-2 \\
t(x)=-2 x^2-1
\end{array}

Find the value of $s(t(3))$.

Understanding the Problem

Before diving into the solution, let's break down what the question is asking. We are given two functions, $s(x)$ and $t(x)$. The function $s(x)$ takes an input $x$, multiplies it by 2, and then subtracts 2. The function $t(x)$ takes an input $x$, squares it, multiplies it by -2, and then subtracts 1. The question asks us to find the value of $s(t(3))$, which means we need to first evaluate $t(3)$, and then use that result as the input for the function $s(x)$. This type of problem involves composite functions, where the output of one function becomes the input of another. Composite functions are a fundamental concept in mathematics, especially in calculus and analysis. Understanding how to evaluate them is crucial for solving more complex problems. This particular problem is a good example of a nested function evaluation, where we have a function within a function. To solve it, we need to work from the inside out. First, we evaluate the inner function, $t(3)$, and then we use the result as the input for the outer function, $s(x)$. This step-by-step approach is essential for dealing with more complex compositions of functions. Remember to pay close attention to the order of operations and the definitions of the functions. By carefully following these steps, we can accurately determine the value of $s(t(3))$.

Step 1: Evaluate $t(3)$

To begin, we need to find the value of $t(3)$. This means we substitute $x = 3$ into the expression for $t(x)$:

$t(x) = -2x^2 - 1$

Substitute $x = 3$:

$t(3) = -2(3)^2 - 1$

Now, we follow the order of operations (PEMDAS/BODMAS). First, we calculate the exponent:

$3^2 = 9$

Next, we multiply by -2:

$-2(9) = -18$

Finally, we subtract 1:

$-18 - 1 = -19$

Therefore, $t(3) = -19$. This result is crucial because it becomes the input for the next step, where we will evaluate the function $s(x)$. The process of substituting the value and following the order of operations is a fundamental skill in algebra. Make sure you understand each step clearly before moving on. This careful approach ensures that we arrive at the correct intermediate result, which is essential for the final answer. By breaking down the problem into smaller, manageable steps, we can avoid errors and gain a better understanding of the overall process of evaluating composite functions. Remember, the value of $t(3)$ is not the final answer; it is simply an intermediate result that we need to use in the next step.

Step 2: Evaluate $s(t(3))$

Now that we have found $t(3) = -19$, we can proceed to find $s(t(3))$, which is the same as $s(-19)$. We substitute $x = -19$ into the expression for $s(x)$:

$s(x) = 2x - 2$

Substitute $x = -19$:

$s(-19) = 2(-19) - 2$

First, we multiply 2 by -19:

$2(-19) = -38$

Next, we subtract 2:

$-38 - 2 = -40$

Therefore, $s(t(3)) = s(-19) = -40$. This is the final answer to the problem. We have successfully evaluated the composite function by working from the inside out. First, we found the value of the inner function, $t(3)$, and then we used that result as the input for the outer function, $s(x)$. This process highlights the importance of understanding the order of operations and the definitions of the functions. By carefully following each step, we were able to arrive at the correct solution. The concept of composite functions is a key topic in mathematics, and this problem provides a good example of how to evaluate them. Remember to always work from the inside out and pay close attention to the order of operations to avoid errors. The final answer, $s(t(3)) = -40$, represents the value of the composite function at the given input.

Final Answer

Thus, the value of $s(t(3))$ is -40.

$s(t(3)) = -40$

In this problem, we encountered the concept of composite functions, where the output of one function serves as the input for another. This is a crucial concept in mathematics, particularly in calculus and analysis. The step-by-step approach we used, evaluating the inner function first and then using its output as the input for the outer function, is a fundamental technique for dealing with composite functions. The functions $s(x)$ and $t(x)$ are examples of simple algebraic functions, but the principle of composition applies to functions of any complexity. Understanding composite functions is essential for tackling more advanced mathematical problems. This example demonstrates how to evaluate a composite function at a specific point, but the concept extends to finding the general expression for a composite function, which involves substituting one function into another in a symbolic way. Mastering the evaluation of composite functions is a valuable skill for any student of mathematics. The process of substitution and simplification is a core skill that is used throughout mathematics and related fields. By practicing problems like this, you can build your confidence and proficiency in working with functions and algebraic expressions.