Problem 56 - Precalculus with Calculus Previews

Question

Find the first three \(x\) -intercepts of the graph of the given function on the positive \(x\) -axis. $$ f(x)=1+\cos \pi x $$

Step-by-Step Solution

Answer

The first three positive x-intercepts are at x = 1, 3, and 5.

Step 1: Understanding the x-intercept

The x-intercept of a function occurs where the value of the function is zero. Therefore, to find the x-intercepts of the function \(f(x) = 1 + \cos \pi x\), we need to set the function equal to zero: \(1 + \cos \pi x = 0\).

Step 2: Solving for x

Start with the equation \(1 + \cos \pi x = 0\). Subtract 1 from both sides to obtain \(\cos \pi x = -1\). The cosine function equals \(-1\) at odd multiples of \(\pi\), so \(\pi x = \pi + 2k\pi\), where \(k\) is an integer.

Step 3: Isolating x

Divide both sides by \(\pi\) to solve for \(x\): \(x = 1 + 2k\) where \(k\) is an integer. This represents the set of x-values where the function has x-intercepts.

Step 4: Finding the positive x-intercepts

Since we need the positive \(x\)-intercepts, substitute the smallest non-negative integers into \(x = 1 + 2k\). For \(k=0\), \(x = 1\); for \(k=1\), \(x = 3\); and for \(k=2\), \(x = 5\). These correspond to the first three positive \(x\)-intercepts.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, defining the x-coordinate of a point on the unit circle. It is periodic, meaning it repeats its values in a regular interval. This interval is called the period.

• The standard cosine function, \(\cos x\), completes one full cycle every \(2\pi\) radians, which is equivalent to 360 degrees.
• It produces values ranging from -1 to 1.
• The cosine of 0 is 1, and it reaches -1 at \(\pi\).

When you modify the argument of the cosine function, as in \(\cos \pi x\), you alter its period. For \(\cos \pi x\), the period becomes 2 instead of \(2\pi\). This change in the period affects where the function crosses the x-axis and other key points like maxima and minima. Understanding how these modifications affect the original cosine function is crucial for solving trigonometric equations and graphing.

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. These equations require you to find angles or other variables that make them true. Solving them often requires an understanding of the properties of trigonometric functions.

• To solve \(1 + \cos \pi x = 0\), the equation is set to find when \(\cos \pi x = -1\).
• This decision is based on knowing that the cosine is -1 at odd multiples of \(\pi\).
• These occur at \((2k + 1)\pi\), which can be easily derived from the periodicity of the cosine function.

In the solution, we rearrange the equation to isolate cosine, then use these properties to determine for which values of \(x\) the equation holds true. Understanding these steps is crucial because trigonometric equations can appear complex without knowledge of these periodic patterns and identities.

Graphing Functions

Graphing functions is a visually useful way to understand relationships between variables in an equation. It helps identify intercepts, intervals of increase or decrease, and other characteristics by plotting the output values against the input.

• The function \(f(x) = 1 + \cos \pi x\) adds 1 to the cosine function, effectively shifting its graph upwards by one unit.
• The x-intercepts, where the function crosses the x-axis, occur when the function value is zero.
• These points are crucial for understanding the function's behavior on the graph.

By setting up the equation to equal zero, and then finding the corresponding values of \(x\) where this occurs, we can easily pinpoint the function's x-intercepts. Identifying these can help predict and understand other features of the graph, such as where it might reach maximum or minimum points, enhancing the overall comprehension of function analysis.