Problem 27 - Calculus

Question

Use a graph of the function to decide whether or not it is invertible. $$f(x)=x^{3}+5 x+10$$

Step-by-Step Solution

Answer

The function \(f(x)=x^{3}+5x+10\) is invertible because it passes the horizontal line test.

Step 1: Understanding Invertibility

A function is invertible if it is one-to-one. This means that for each output from the function, there is exactly one input that produces it. A graph can help us determine this by checking if any horizontal line intersects the graph at more than one point.

Step 2: Review Horizontal Line Test

The horizontal line test states that if any horizontal line intersects the graph of the function at most once, then the function is one-to-one and thus invertible. We will apply this test to the given function's graph.

Step 3: Graph Construction

Plot the graph of the function \( f(x)=x^{3}+5x+10 \). This can be done using graphing software or manually by plugging in various values of \(x\) to get corresponding \(f(x)\) values, and then connecting these points smoothly, noting that it's a polynomial of degree 3.

Step 4: Analyze the Graph

Examine the plotted graph to see how the curve behaves. Generally, a cubic function graph has an 'S' shape. Check whether any horizontal line would intersect the graph more than once.

Step 5: Apply Horizontal Line Test

Draw horizontal lines across the graph of \(f(x)=x^{3}+5x+10\). Observe that, due to its continuous increase and absence of repeated values for any \(y\), each horizontal line intersects the graph at exactly one point.

Key Concepts

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one and therefore invertible. To apply this test, draw a series of horizontal lines across the graph of a function:

• If any horizontal line crosses the graph more than once, the function fails the test and is not one-to-one.
• If each horizontal line crosses the graph at most once, the function passes the test and is one-to-one.

In the case of the function \( f(x)=x^3+5x+10 \), you would observe whether any horizontal line intersects the graph more than once. Since this function is cubic, it generally forms an 'S' shaped curve. If no horizontal line intersects the curve more than once, as is the case here, the function is one-to-one and hence invertible.This test is particularly useful because it gives a quick visual confirmation of a function's behavior regarding invertibility, without needing to calculate inverse functions for complex graphs.

One-to-One Function

A one-to-one function, often denoted as injective, is a function where each output value is produced by exactly one input value. One-to-one functions have a unique property: each point in the output corresponds to only one input value. For a function to be invertible, it must be one-to-one:

• This means no value in the range is the result of more than one input value.
• If a horizontal line drawn across the graph of a function touches the curve at more than one point, this indicates that the function is not one-to-one.

In algebraic terms, a function \( f \) is one-to-one if, for any two different inputs \( x_1 \) and \( x_2 \), the outputs are also different: \( f(x_1) eq f(x_2) \). In the case of our cubic function \( f(x)=x^3+5x+10 \), it passes the horizontal line test, establishing it as a one-to-one function. Thus, we can say it is invertible, allowing us to find its inverse with confidence.

Cubic Functions

Cubic functions are polynomial functions of degree three, which can be written in the standard form \( f(x) = ax^3 + bx^2 + cx + d \). These functions have distinct characteristics that make them interesting and useful:

• The graph of a cubic function often has an 'S' shape that passes through the origin or is offset, depending on the coefficients.
• They can have one, two, or three real roots, which are the x-intercepts where the graph crosses the x-axis.

For our given function \( f(x)=x^3+5x+10 \), there is no \( x^2 \) term which simplifies the analysis somewhat. When graphing this function, you observe the typical 'S' shape. The function shows a continuous increase or decrease without repeating \( y \) values in any segment, crucial for determining invertibility through the horizontal line test.Cubic functions are versatile, appearing often in real-world applications such as physics and engineering, often modeling phenomena with an inherent curve rather than a direct linear relation.