Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) for which each function is continuous. \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x \leq 1 \\ 2 x-1 & \text { if } x>1\end{array}\right.\)

Step-by-Step Solution

Answer

So, the function \(f(x)\) is continuous for all values of \(x\), as both individual functions are continuous in their respective regions, and the piecewise function is continuous at the transition point (\(x = 1\)).

Step 1: Check Continuity for the Individual Functions

We need to check the continuity of \(f(x) = x\) for \(x \leq 1\) and \(f(x) = 2x - 1\) for \(x > 1\). Since both of these are linear functions (polynomials of degree 1), they are continuous for all values of \(x\). Therefore, they are continuous in their respective specified regions.

Step 2: Check Continuity at the Transition Point

Now, we need to check the continuity of the overall piecewise function, specifically at the transition point, \(x = 1\). To do this, we need to make sure that both cases approach the same value when \(x = 1\), that is the left-hand limit equals the right-hand limit. For the left-hand limit, when \(x \leq 1\): \[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x = 1 \] For the right-hand limit, when \(x > 1\): \[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x - 1) = 2(1) - 1 = 1 \] Since both the left-hand limit and the right-hand limit are equal (both are 1), the overall piecewise function is continuous at the transition point, \(x = 1\).

Step 3: Conclusion

Since both the individual functions are continuous in their respective regions and the overall piecewise function is continuous at the transition point (\(x = 1\)), we can conclude that \(f(x)\) is continuous for all values of \(x\).

Key Concepts

Limit of a Function

Understanding the limit of a function is like finding what value the function approaches as the input gets closer to a certain point. It's not necessarily the value the function actually reaches, but the value it gets infinitely close to. For example, as you walk towards a wall, your distance to the wall is the 'limit'; you never actually become the wall, but you get very close.

With mathematics, we denote this idea using the symbol \( \lim \) followed by the function and the point we're inching towards. If both the left and right limits as we approach a certain point are the same, then the function has a limit at that point. In the exercise, we examined the limits from both directions at \( x = 1 \) and found they both equaled 1, so \( f(x) \) has a limit at \( x = 1 \).

Piecewise-Defined Functions

Think of piecewise-defined functions like a choose-your-own-adventure book: depending on where you are in the story (or which \( x \) value you're working with), you follow different rules (or functions). These types of functions are split into pieces, each with its own equation that applies to certain intervals of \( x \).

To analyze their continuity, you must look at each 'piece' individually and then examine the points where the pieces meet—these are your 'cliffhangers' that keep the adventure going smoothly. In the given exercise, the function \( f(x) \) has two pieces, one for \( x \leq 1 \) and another for \( x > 1 \) and continuity was determined for both regions.

Continuity at a Point

Imagine you're drawing a line without lifting your pencil off the paper—this line represents a continuous function. Continuity at a point means that you can draw right through that point without any jumps or interruptions. Mathematically, for a point to be continuous, three conditions must be met: the function is defined at that point; the limit of the function as it approaches the point exists; and the function's value at that point is equal to the limit.

In our exercise, we checked the value and the limits from both sides at \( x = 1 \) and found them to be equal, which meant the function was continuous at that point. There was no 'lifting the pencil off the paper' at \( x = 1 \).

Linear Functions

Like the steady incline or decline of a hill, linear functions are straight lines where you can calculate the slope (the steepness) and the y-intercept (where it crosses the y-axis). The general formula is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These are the simplest types of functions and are continuous everywhere—no breaks, jumps, or sudden changes.

In our problem, both pieces of \( f(x) \) are linear functions—one with a slope of 1 (simply \( x \) itself) and the other with a slope of 2 (the equation \( 2x - 1 \)). Since linear functions are always continuous, each piece of our piecewise function had no continuity issues individually.