Question

ABSTRACT REASONING Suppose \(f(x) \rightarrow \infty\) as \(x \rightarrow-\infty\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow \infty\). Describe the end behavior of \(g(x)=-f(x)\). Justify your answer.

Short answer

The end behavior of \(g(x) = -f(x)\) is that it tends to negative infinity as \(x\) approaches negative infinity and to positive infinity as \(x\) approaches positive infinity.

Step-by-step solution

Understand the given function \(f(x)\)

It's given that as \(x \rightarrow -\infty, f(x) \rightarrow \infty\), and when \(x \rightarrow \infty, f(x) \rightarrow -\infty\). This describes the end behavior of the function \(f(x)\). As \(x\) approaches negative infinity, \(f(x)\) increases without bound, and as \(x\) advances towards positive infinity, \(f(x)\) decreases without bound.

Describe the function \(g(x)\)

Now implement the function \(g(x) = -f(x)\). This is simply the original function \(f(x)\) multiplied by -1. So, we are flipping the function in the x-axis.

Derive the end behavior of \(g(x)\)

Applying the property to \(g(x)\) that multiplying a function by -1 inverts its values, we get: as \(x \rightarrow -\infty, g(x) = -f(x) \rightarrow -\infty\) (since \(f(x)\) was tending to \(\infty\)). Similarly as \(x \rightarrow \infty, g(x) = -f(x) \rightarrow \infty\) (since \(f(x)\) was tending towards \(-\infty\)).

Key concepts

Function Transformation

When discussing function transformation, we often refer to operations that modify the appearance of a graph. One of the most common transformations is reflection, which can be applied by multiplying the function by -1.
This operation has a significant impact on the graph of the function. In the case of the function \( g(x) = -f(x) \) derived from \( f(x) \), this reflects the function across the x-axis.
Here's how it works:

• If a point on the original function is above the x-axis, its reflected image on \( g(x) \) will appear directly below the x-axis.
• If a point is below the x-axis, its image will be directly above.

This reflection affects the entire graph and is essential in determining how the function's end behavior changes.

Infinity Limits

Infinity limits help us understand the behavior of functions as they approach infinite values or shrink towards large negative numbers.
In the context of our exercise, taking the original function \( f(x) \):

• As \( x \to -\infty \), \( f(x) \to \infty \), which means that as \( x \) gets very negative, \( f(x) \) increases without bound.
• Conversely, as \( x \to \infty \), \( f(x) \to -\infty \), indicating that \( f(x) \) decreases without bound as \( x \) becomes very positive.

When we transform \( f(x) \) to \( g(x) = -f(x) \), the limits are inverted:

• For \( x \to -\infty \), \( g(x) \to -\infty \).
• For \( x \to \infty \), \( g(x) \to \infty \).

This demonstrates how transformations like reflections can change the end behavior as \( x \) moves toward positive or negative infinity.

Graph Behavior

Understanding graph behavior includes studying how a function behaves as its input values become extremely large or small. This is especially noticeable at the edges, or ends, of the graph.
For the original function \( f(x) \), it was pointed out that:

• It shoots upwards as \( x \) heads left towards negative infinity, conveying an increasing trend.
• It drops downwards as \( x \) moves right towards positive infinity, showing a decreasing pattern.

This characteristic is altered when the function is transformed to \( g(x) = -f(x) \).
The new behavior becomes:

• As \( x \rightarrow -\infty \), \( g(x) \) will instead drop downwards.
• And as \( x \rightarrow \infty \), \( g(x) \) heads upwards.

This exploration of graph behavior helps determine what to expect when graphing or analyzing functions like \( g(x) = -f(x) \), providing critical insight into the function's nature at extreme values.