Determine the value of a for a given function.

In summary, in order to determine the value of "a" for the given function, you must solve for "a" such that the left-side limit as x approaches 2 is equal to 8. This can be done by setting the left-side limit as 2a+7 and the right-side limit as 8-a, and solving for "a". The solution is a = 1/2.
  • #1
Torshi
118
0
Determine the value of "a" for a given function.

Homework Statement



Consider the following function: f(x) { ax+7, x<2 and 4x-a, x≥2

A.) Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8. You mus demonstrate all limits.

Homework Equations



ax+7, x<2 and 4x-a, x≥2

The Attempt at a Solution



Approaching from left I got lim x->2- (2a+7) and from right lim x->2+ (8-a)
Am I trying to find a? If I set both limits equal I get a = .33, but the question asks determine the value of such that lim x->2- f(x) = 8 ? That would mean a =1/2 because 2* 1/2 = 1 + 7 = 8, but I don't think that's right unless I'm reading the question wrong..


Edit:
Or do I just state that ax+7 since for this question says lim x->2- (approaching) left f(x) = 8 do I say coming from the left is 8 while from the right it's still 8-a then set those two equal to calculate for "a"? Getting -1 as my answer for "a"?
 
Last edited:
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  • #2


Torshi said:

Homework Statement



Consider the following function: f(x) { ax+7, x<2 and 4x-a, x≥2

A.) Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8. You mus demonstrate all limits.

Homework Equations



ax+7, x<2 and 4x-a, x≥2

The Attempt at a Solution



Approaching from left I got lim x->2- (2a+7) and from right lim x->2+ (8-a)
This is what you mean:
$$ \lim_{x \to 2^-}f(x) = 2a + 7$$
$$ \lim_{x \to 2^+}f(x) = 8 - a$$
Torshi said:
Am I trying to find a?
Yes, so that the left-sided limit as x approaches 2 equals 8.
Torshi said:
If I set both limits equal I get a = .33, but the question asks determine the value of such that lim x->2- f(x) = 8 ? That would mean a =1/2 because 2* 1/2 = 1 + 7 = 8
What are you saying here? 2 * 1/2 = 1 ≠ 8
Torshi said:
, but I don't think that's right unless I'm reading the question wrong..


Edit:
Or do I just state that ax+7 since for this question says lim x->2- (approaching) left f(x) = 8 do I say coming from the left is 8 while from the right it's still 8-a then set those two equal to calculate for "a"? Getting -1 as my answer for "a"?
 
  • #3


Mark44 said:
This is what you mean:
$$ \lim_{x \to 2^-}f(x) = 2a + 7$$
$$ \lim_{x \to 2^+}f(x) = 8 - a$$
Yes, so that the left-sided limit as x approaches 2 equals 8.

Alright, then now I'm assuming set 8 = 8-a
 
  • #4


Torshi said:
Alright, then now I'm assuming set 8 = 8-a --> which equals a = -1. Thanks
No, for two reasons.

1) You are supposed to
Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8
2)If 8 = 8 - a, then a = -1 is NOT the solution.
 
  • #5


For the left-side limit, a = 1/2.
You had that, but wrote something that didn't make sense.

If a = 1/2, then 2a + 7 = 2(1/2) + 7 = 8
 
  • #6


Mark44 said:
No, for two reasons.

1) You are supposed to
2)If 8 = 8 - a, then a = -1 is NOT the solution.
Sorry, i realized that I switched it to a-8 for some odd reason. Nevermind, I don't know how I came up with that. My fault.

8 = 8 - a
a+8 = 8
a = 0
?
 
  • #7


The question is asking about the left-side limit, not the right-side limit.
 
  • #8


Mark44 said:
The question is asking about the left-side limit, not the right-side limit.

Wait, but I thought the question said to determine the value of "a" such that lim x->2- f(x) = 8 meaning that approaching from the left lim x->2- = 8 and from the right that being 8-a, we solve for "a" which is from lim x->2+ (8-a) , so 8 = 8-aif it's the other way around a = 1/2
 
Last edited:
  • #9


I think I get it.

2a+7 = 8 then solve for a

a = 1/2

Thanks
 
  • #10


Mark44 said:
For the left-side limit, a = 1/2.
You had that, but wrote something that didn't make sense.

If a = 1/2, then 2a + 7 = 2(1/2) + 7 = 8

I did say that, but I said it without solving so it was just an assumption by looking at the problem, the reason why I didn't go with it because it sounded too easy even though it made sense. My fault. Thank you.
 

Related to Determine the value of a for a given function.

1. What is the purpose of determining the value of a for a given function?

The value of a in a function represents the coefficient or constant that affects the shape and position of the graph. It allows us to make predictions and understand the behavior of the function.

2. How do you determine the value of a for a linear function?

For a linear function in the form y = mx + b, a represents the slope or rate of change. It can be calculated by dividing the change in y values by the corresponding change in x values between two points on the graph.

3. What is the significance of determining the value of a for a quadratic function?

In a quadratic function, a represents the coefficient of the squared term. This value determines the shape of the parabola and can give information about the maximum or minimum point of the function.

4. Can the value of a for a given function change?

Yes, the value of a can vary depending on the specific function and its parameters. It can also be manipulated by applying transformations to the original function.

5. How can determining the value of a for a given function be applied in real-world situations?

In many fields of science, such as physics and engineering, functions are used to model real-world phenomena. By determining the value of a, we can make predictions and solve problems related to these phenomena.

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