Problem 26 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\left\\{\begin{array}{cl} \frac{1}{x} & \text { for } x \geq 1 \\ 2 x+c & \text { for } x<1 \end{array}\right. $$ (a) Graph \(f(x)\) when \(c=0\), and determine whether \(f(x)\) is continuous for this choice of \(c\). \begin{array}{l} \text { (b) How must you choose } c \text { so that } f(x) \text { is continuous for all }\\\ x \in(-\infty, \infty) ? \end{array}

Step-by-Step Solution

Answer

For continuity, choose \(c = -1\). For \(c=0\), the function is not continuous at \(x=1\).

Step 1: Understand the piecewise function

The function \(f(x)\) is defined as a piecewise function with two parts: - For \(x \geq 1\), \(f(x) = \frac{1}{x}\).- For \(x < 1\), \(f(x) = 2x + c\).We want to examine and graph the function, considering continuity for different values of \(c\).

Step 2: Graph the function for c=0

For \(c=0\), the function simplifies to: - \(f(x) = \frac{1}{x}\) for \(x \geq 1\).- \(f(x) = 2x\) for \(x < 1\).To graph: 1. For \(x \geq 1\), plot \(f(x) = \frac{1}{x}\). This curve is the hyperbola that gets close to the x-axis as it moves to the right and approaches 1 as \(x\) gets closer to 1 from the right.2. For \(x < 1\), plot \(f(x) = 2x\), a straight line passing through the origin with a slope of 2.

Step 3: Check continuity at x=1 for c=0

A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are equal.For \(x=1\):- Right-hand limit (\(x \to 1^+\)) is \(f(x) = \frac{1}{x} = 1\).- Left-hand limit (\(x \to 1^-\)) is \(f(x) = 2(1) + 0 = 2\).- \(f(1)\) is defined on \(x \geq 1\) side, so \(f(1) = 1\).Since left-hand limit, right-hand limit, and \(f(1)\) do not match, \(f(x)\) is not continuous at \(x=1\) for \(c=0\).

Step 4: Determine c for continuity for all x

For \(f(x)\) to be continuous for all \(x\), particularly at \(x=1\), the left-hand and right-hand limits as \(x\) approaches 1 must be the same.- Left-hand limit (\(x \to 1^-\)) needs to be \(2(1) + c = 2 + c\).- Right-hand limit (\(x \to 1^+\)) is still \(1\).Setting \(2 + c = 1\), we solve for \(c\):\[2 + c = 1\] \[c = 1 - 2\] \[c = -1\]Choosing \(c = -1\), the limits match and \(f(x)\) is continuous for all \(x\).

Key Concepts

Graphing Functions

Graphing a piecewise function can initially seem tricky, but breaking it into manageable parts helps.
To understand the function graphically, you need to note each piece's unique behavior and domain.
For the given function, when graphed:

• For the segment where \(x \geq 1\), the expression \(f(x) = \frac{1}{x}\) forms a hyperbola. This graph curves downward as \(x\) increases and approaches 1 from the right. It never touches the x-axis but gets closer to it, indicating it's an asymptote.
• For the section where \(x < 1\), the function becomes \(f(x) = 2x\) for \(c = 0\). This is a straightforward linear equation with a slope of 2, so you'd see a straight line at an angle, passing through the origin.

Graphing each part separately first can clarify how the overall function behaves. Then, you locate any discontinuities or where these segments join, such as around \(x = 1\). This will help you assess if the graph smoothly connects across its entire domain.

Continuity of Functions

The concept of continuity is significant when working with piecewise functions. A function is continuous at a point if it doesn't "break" at that point, meaning the graph doesn't jump or have a hole.
For the function \( f(x) \) in this problem:

• Evaluate the function's continuity where the different pieces meet, particularly at \(x = 1\).
• A continuous function must satisfy:
  • The limit from the left \(\lim_{{x \to 1^-}} f(x)\) matches the limit from the right \(\lim_{{x \to 1^+}} f(x)\).
  • The combined limits equal the actual function value at the point \(f(1)\).
• In this case, when \(c=0\), \(f(x)\) isn't continuous because the different function parts at \(x = 1\) diverge.
• Choosing \(c = -1\) fixes this issue, as the function becomes continuous, offering a seamless graph from \(-\infty\) to \(\infty\).

Understanding continuity helps identify why and where a function may "break," guiding how to manipulate the equation to resolve these discontinuities.

Limits in Calculus

Limits play a pivotal role in understanding function behavior, especially around potential discontinuities. They help us anticipate what happens to a function's value as it inches closer to a particular point from either side.
For the given piecewise function:

• The right-hand limit as \(x\) approaches 1 from the right is computed using \(f(x) = \frac{1}{x}\), making the right-hand limit equal 1.
• The left-hand limit approaching 1 from the left direction uses \(f(x) = 2x + c\). This results in \(2 \times 1 + c\) when \(x = 1\).
• To make the function continuous, the left and right limits need to synchronize: setting \(2 + c = 1\) leads to \(c = -1\).

By mastering limits, you can predict a function's behavior at points without direct substitution, vital for graphs with holes or jumps. This ensures that even when a function changes dramatically, the transition between pieces is smooth and predictable.