Question

Find the values of \(a\) and \(b\) so that the following function is continuous everywhere. $$ f(x)=\left\\{\begin{array}{ll} x+1 & \text { if } x<1 \\ a x+b & \text { if } 1 \leq x<2 \\ 3 x & \text { if } x \geq 2 \end{array}\right. $$

Short answer

The values are \( a = 4 \) and \( b = -2 \).

Step-by-step solution

Understand Continuity Conditions

A function is continuous at a point if the following three conditions are satisfied: 1) The function is defined at that point. 2) The limit of the function as it approaches the point from both sides exists. 3) The limit of the function as it approaches the point equals the function's value at that point. For this piecewise function, check the points where the definition changes, i.e., at \( x = 1 \) and \( x = 2 \).

Check Continuity at \( x = 1 \)

At \( x = 1 \), we have:- From the left, \( f(x) = x + 1 \), thus \( f(1^{-}) = 1 + 1 = 2 \).- From the function, \( f(x) = a \cdot 1 + b = a + b \) for \( x \geq 1 \), so we require \( f(1) = a + b \).- For continuity, set \( f(1^{-}) = f(1) \), so \( a + b = 2 \).

Check Continuity at \( x = 2 \)

At \( x = 2 \), we have:- From the quadratic function, \( f(x) = a \, x + b \), thus \( f(2^{-}) = 2a + b \).- From the right, \( f(x) = 3x \), so \( f(2) = 3 \times 2 = 6 \).- For continuity, set \( 2a + b = 6 \).

Solve System of Equations

We have two equations from the continuity conditions:1. \( a + b = 2 \)2. \( 2a + b = 6 \)Subtract the first equation from the second to find \( a \):\( (2a + b) - (a + b) = 6 - 2 \)\( a = 4 \).Substitute \( a = 4 \) into the first equation to find \( b \):\( 4 + b = 2 \)\( b = -2 \).

Verify Solutions

Verify the solutions by substituting \( a = 4 \) and \( b = -2 \):- At \( x = 1 \): \( f(1) = 4 \times 1 - 2 = 2 \), matches \( f(1^{-}) = 2 \).- At \( x = 2 \): \( f(2) = 4 \times 2 - 2 = 6 \), matches \( f(2^{+}) = 6 \).Both points satisfy the continuity conditions, confirming the solution is correct.

Key concepts

Continuous Functions

A continuous function can be thought of as a seamless and unbroken curve, without any abrupt jumps or gaps. To determine if a function is continuous, we check specific conditions.

• The function must be defined at a point, meaning the function provides a concrete output or value at that point.
• The limit of the function must exist as it approaches the point from either side of the x-axis.
• The actual value of the function at the point must equal the limit value that the function approaches.

When these conditions are met at every point in its domain, we can confidently say that the function is continuous. In the exercise provided, we aim to ensure the function is continuous everywhere, especially at the transition points where the function's expression changes. This exercise involves figuring out suitable values for the parameters to keep the function smooth throughout its domain.

Limits and Continuity

Limits serve as the backbone of understanding continuity. They tell us what value a function creeps toward as the input comes closer to a point from either side.
Consider the idea of traveling towards a destination. Even if you're not there yet, knowing you're steadily approaching tells a lot about where the journey leads.
In the context of functions, continuity requires that the limit matches the actual function value at the point in question.
In the example problem, we identify key areas (such as at the points where the rule of the piecewise function changes) and check these limits:

• At \(x = 1\), the limit as we approach from the left ((x)=x+1")) equals 2.
• At \(x = 1\), the limit approaching from the right should equal \(f(x)=ax+b\), which we set to 2 to maintain continuity.
• At \(x = 2\), the limit from the left ((x)=ax+b")) needs to match the limit from the right at 6.

Effectively using limits helps in solving for the unknowns that make functions adapt smoothly across different sections.

Piecewise Function Analysis

Piecewise functions can appear challenging because they consist of different expressions used over various parts of the domain. Analyzing these functions involves ensuring consistency at the junctions where the pieces meet.
To do this, we analyze the behavior at each segment's start and end.

• The piecewise function in the exercise has three parts. \(x+1\) when \(x < 1\), \(ax + b\) when \(1 \leq x < 2\), and \(3x\) when \(x \geq 2\).
• Our task was to find values of \(a\) and \(b\) that make the entire function continuous.
• At \(x = 1\) and \(x = 2\), setting equations of limits from adjoining segments equal to each other reveals \(a\) and \(b\).

By solving these equations, we keep the function graphs lining up perfectly, avoiding any jumps or breaks between the pieces. This analysis leads to discovering the seamless transition points, ensuring the piecewise function maintains continuity throughout its domain.