Problem 24 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$g(x)=-2 x$$

Step-by-Step Solution

Answer

As \(x\) approaches positive infinity (\(+\infty\)), \(g(x)\) approaches negative infinity (\(-\infty\)). As \(x\) approaches negative infinity (\(-\infty\)), \(g(x)\) approaches positive infinity (\(+\infty\)).

Step 1: Behavior as \(x\) approaches positive infinity

Substitute \(+\infty\) into the function \(g(x) = -2x\). We can observe that as \(x\) increases to \(\infty\), \(g(x)\) will decrease to \(-\infty\). This is because a positive value multiplied by negative two yields a negative value, and as the positive value (x) increases, the product grows more negative.

Step 2: Behavior as \(x\) approaches negative infinity

Next, substitute \(-\infty\) into the function \(g(x) = -2x\). We can see that as \(x\) decreases to \(-\infty\), \(g(x)\) will increase to \(+\infty\). This occurs because a negative value multiplied by negative two results in a positive value, and as the negative value (x) decreases, the positive product becomes larger.

Key Concepts

Linear Functions

Linear functions are the simplest form of algebraic functions. They have the general form of f(x) = mx + b, where m represents the slope and b is the y-intercept—the point where the line crosses the y-axis. In our exercise g(x) = -2x, this is a linear function without a y-intercept (b=0), and with a slope of m = -2. The slope determines the steepness of the line and the direction it tilts; since our slope is negative, our line tilts downwards as we move from left to right.

One of the defining characteristics of linear functions is their predictability. No matter how large or small the value of x becomes, the output of the function is directly proportional to x. This means that if you double the value of x, the value of g(x) simply doubles—but in the opposite direction, because of the negative slope.

Limits at Infinity

The concept of limits at infinity helps us understand the behavior of functions as the input, or x, gets very large or very large negative values. Mathematically, we express this as x approaches infinity (\(\infty\)) or negative infinity (\(-\infty\)). It essentially tells us where the function is 'heading' or what value it is approaching, even though x may never actually reach infinity. For our function g(x) = -2x, when analyzing the limit as x approaches infinity, we notice that the function decreases without bound and similarly, as x approaches negative infinity, the function increases without bound.

Considering these limits does not necessarily give us a numerical value but rather a behavior or trend. It's an important aspect to grasp because it forms the foundation for understanding the global behavior of functions, especially for evaluating the behavior of functions at extreme values of x which cannot be plainly computed.

Negative Infinity Behavior

When we speak of a function's behavior at negative infinity, we are focusing on what happens to the value of the function, g(x), as x gets very large in the negative direction (\(x \to -\infty\)). It's important to consider that this is a hypothetical scenario; x can decrease without limit, and we're interested in the trend rather than a specific value. For the given linear function g(x) = -2x, since we have a negative slope, as x becomes a large negative number, the multiplication by -2 leads to a large positive result. Therefore, the end behavior as x approaches negative infinity is that g(x) 'heads' towards positive infinity (\(+\infty\)).

This negative infinity behavior can be used to determine how a graph will appear when x is very small (in the negative sense). In graphs, this behavior dictates where the 'tail' of the graph will be positioned: either up towards positive infinity or down towards negative infinity. In our case, the 'tail' of g(x) points upwards as x extends to the left.