Evaluate (mn)(x) At X=-3 With N(x)=x

In the realm of mathematics, functions are fundamental building blocks that allow us to describe relationships between variables. When we encounter expressions like $(mn)(x)$, we're delving into the world of function multiplication. This operation, while seemingly straightforward, has specific rules and applications that are crucial for solving various mathematical problems. Let's break down what $(mn)(x)$ means and how to evaluate it, particularly when $n(x) = x$ and we need to find the value at $x = -3$.

Understanding Function Multiplication

The notation $(mn)(x)$ signifies the product of two functions, $m(x)$ and $n(x)$. Mathematically, this is defined as:

$$(mn)(x) = m(x) imes n(x)$$

In essence, to find the value of the product of two functions at a specific point $x$, you first find the value of each individual function at that point and then multiply those values together. This concept is akin to multiplying two numbers; you find each number's value and then perform the multiplication.

The Specific Case: $n(x) = x$

Now, let's introduce the specific function given in our problem: $n(x) = x$. This is one of the simplest functions, known as the identity function. For any input value, the identity function outputs that same value. So, if $x$ is 5, $n(x)$ is 5. If $x$ is -10, $n(x)$ is -10, and so on. This characteristic makes it a very useful function in many mathematical contexts.

When $n(x) = x$, our function multiplication expression becomes:

$$(mn)(x) = m(x) imes x$$

This means that the product of function $m$ and the identity function $n$ at any point $x$ is simply the function $m(x)$ multiplied by $x$. This simplifies the problem considerably, as we already know the behavior of $n(x)$.

Evaluating $(mn)(x)$ at $x = -3$

We are asked to evaluate $(mn)(x)$ specifically when $x = -3$. Using our definition of function multiplication and the given function $n(x) = x$, we substitute $-3$ for $x$ into the expression:

$$(mn)(-3) = m(-3) imes n(-3)$$

Since we know that $n(x) = x$, we can directly determine the value of $n(-3)$:

$$n(-3) = -3$$

Now, our equation becomes:

$$(mn)(-3) = m(-3) imes (-3)$$

At this stage, to find the final numerical answer, we would need the definition of the function $m(x)$. However, the problem statement implies that $m(x)$ is a function that is either provided implicitly or is meant to be considered in a general sense. If $m(x)$ were, for instance, $m(x) = 2x + 1$, then $m(-3) = 2(-3) + 1 = -6 + 1 = -5$. In that hypothetical case, $(mn)(-3) = (-5) imes (-3) = 15$.

Crucially, the problem as stated has a missing piece of information: the definition of the function $m(x)$. Without knowing what $m(x)$ is, we cannot compute a specific numerical value for $(mn)(-3)$. The problem asks for a numerical answer in the format (mn)(-3)=□, which suggests that either $m(x)$ is meant to be a universally understood function in this context (which is unlikely without further definition), or there's an assumption missing from the prompt.

Let's assume, for the sake of providing a complete example structure, that the problem intended to provide $m(x)$. If, for example, $m(x)$ was simply $m(x) = 5$ (a constant function), then:

$$(mn)(-3) = m(-3) imes n(-3)$$

$$= 5 imes (-3)$$

$$= -15$$

If, on the other hand, $m(x)$ was $m(x) = x^2$, then:

$$(mn)(-3) = m(-3) imes n(-3)$$

$$= (-3)^2 imes (-3)$$

$$= 9 imes (-3)$$

$$= -27$$

Given that the prompt has a placeholder for a specific numerical answer, it's highly probable that the function $m(x)$ was intended to be provided or is understood from a prior context not included here. For instance, if $m(x)$ was the function such that $(mn)(x)$ was a specific known function, or if $m(x)$ was defined in a preceding problem. Without $m(x)$, the expression $(mn)(-3)$ can only be simplified to $m(-3) imes (-3)$.

Revisiting the Problem Structure

The question asks to