Find a and b for Continuous Function on Real Line

In summary, the constants a and b for the given function to be continuous on the entire real line are a = 1 and b = -1. These values satisfy the two linear equations -a + b = 2 and 3a + b = -2, ensuring that the left-side and right-side limits are equal at x = -1 and x = 3. Checking the answer by substituting these values into the original function confirms its continuity.
  • #1
fr33pl4gu3
82
0
Q.

Determine the constants a and b so that the function is continuous on the entire real line.

2 if x <= -1
f(x) = ax+b if -1 < x < 3
-2 if x >= 3

Ans:

a = 1; b = -1

I wonder if the answer is right??
 
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  • #2
fr33pl4gu3 said:
a = 1; b = -1

I wonder if the answer is right??

Almost … check it again … :wink:
 
  • #3
Which one is wrong, a or b, or both??
 
  • #4
Suppose your answer IS correct, that would mean

2 if x <= -1
f(x) = x - 1 if -1 < x < 3
-2 if x >= 3

However when x = -1, you get 2 (from the criterion x <= -1) and -2 for the criterion -1 < x < 3. Also when x = 3, you get 2 for the criterion -1 < x < 3 and -2 for the criterion x >= -3. You don't want those jumps if your f is cont.
 
  • #5
Exactly. You shouldn't have to "wonder" if your answer is correct, you can check it yourself. Is the left-side limit of f(x) equal to the right-side limit at x=-1? at x = 3?
 
  • #6
you have 2 linear equations with 2 unknowns

-a + b = 2
3a + b = -2
 

FAQ: Find a and b for Continuous Function on Real Line

1. What is the definition of a continuous function on the real line?

A continuous function on the real line is a function where the graph is an unbroken curve with no gaps or jumps. In other words, it has a smooth and continuous graph without any breaks or holes.

2. How do you find the values of a and b for a continuous function on the real line?

To find the values of a and b for a continuous function on the real line, you can use the slope-intercept form of a line, y = mx + b. The value of a represents the slope of the line, while b represents the y-intercept (where the line crosses the y-axis). You can use this information to determine the equation of the continuous function.

3. What is the importance of finding a and b for a continuous function on the real line?

Finding a and b for a continuous function on the real line is important because it helps us understand the behavior of the function. The value of a tells us how steep the line is, while the value of b tells us where the line intersects the y-axis. This information can help us make predictions and solve problems involving the function.

4. Can a continuous function on the real line have more than one value for a and b?

Yes, a continuous function on the real line can have more than one value for a and b. This is because there are many different lines that can have the same slope and y-intercept, and still be considered continuous functions on the real line. However, the values of a and b will determine the specific equation for the function.

5. How can you graph a continuous function on the real line given the values of a and b?

To graph a continuous function on the real line, you can use the values of a and b to determine the slope and y-intercept. From there, you can plot points on the graph and connect them with a smooth line. Alternatively, you can also use the slope-intercept form of a line to create a table of values and plot points from there.

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