Two questions: Functions

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Vertciel

Junior Member
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Hello everyone,

I would appreciate any help for the following question and problem!

Thank you very much.

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1. I just want to confirm that the following algebra is "legal" in math. If \(\displaystyle 3f(x) = 18\), can I divide both sides to get \(\displaystyle f(x) = 6\)? In other words, can I treat the function notation as a variable like x and ?

2. What is the minimum value of the function \(\displaystyle f(x) = 2^{x^2 - 2x}\)?

I am confused by the power of \(\displaystyle x^2 - 2x\). I know that the exponent can be factored to \(\displaystyle f(x) = 2^{x(x-2)}\), but would this help at all?
 
Vertciel said:
1. I just want to confirm that the following algebra is "legal" in math. If \(\displaystyle 3f(x) = 18\), can I divide both sides to get \(\displaystyle f(x) = 6\)? In other words, can I treat the function notation as a variable like x and ?

It is perfectly legal - f(x) is just an entity (a stuff).

2. What is the minimum value of the function \(\displaystyle f(x) = 2^{x^2 - 2x}\)?

I am confused by the power of \(\displaystyle x^2 - 2x\). I know that the exponent can be factored to \(\displaystyle f(x) = 2^{x(x-2)}\), but would this help at all?

What is the first condition that need to be satisfied for a function to have a local minimum or maximum? (have you done differentiation?)
Click to expand...
 
Subhotosh Khan said:
Vertciel said:
2. What is the minimum value of the function \(\displaystyle f(x) = 2^{x^2 - 2x}\)?

I am confused by the power of \(\displaystyle x^2 - 2x\). I know that the exponent can be factored to \(\displaystyle f(x) = 2^{x(x-2)}\), but would this help at all?

What is the first condition that need to be satisfied for a function to have a local minimum or maximum? (have you done differentiation?)
Click to expand...
Click to expand...
Thanks Subhotosh Khan for your reply.

For 2, a function would need to have a vertex to have a minimum or maximum value. In this case, \(\displaystyle f(x) = 2^{x^2 - 2x}\) would be a quadratic function. Unfortunately, we haven't done differentiation; I am in a precalculus math course in high school.
 
Are you sure the problem (#2) is not:

\(\displaystyle f(x) = 2x^2 - 2x\)

The problem you posted will be very difficult to do without calculus.
 
Actually, to find the minimum value, we can look at the quadratic in the exponent.

Use -b/(2a). \(\displaystyle x^{2}-2x\), b=-2, a=1

-(-2)/(2(1))=1

The minimum is at 1.
 
Vertciel said:
For 2, a function would need to have a vertex to have a minimum or maximum value. In this case, \(\displaystyle f(x) = 2^{x^2 - 2x}\) would be a quadratic function. Unfortunately, we haven't done differentiation; I am in a precalculus math course in high school.
Click to expand...
So use what you've learned in algebra. The function, being 2 to some power, will be smallest when the power is at its lowest value. So what is the minimum of the quadratic g(x) = x[sup:2y5tje69]2[/sup:2y5tje69] - 2x?

(Hint: Find the vertex!) :wink:

Eliz.
 
Thank you very much to everyone who replied.

The minimum of the \(\displaystyle g(x) = x^2 - 2x\) is 1, but the answer given at the back of the textbook is \(\displaystyle \frac{1}{2}\). Can anyone please explain the difference?
 
Vertciel said:
The minimum of the \(\displaystyle g(x) = x^2 - 2x\) is 1, but the answer given at the back of the textbook is \(\displaystyle \frac{1}{2}\). Can anyone please explain the difference?
Click to expand...
You need to x-value at which you get the minimum of the exponent. Plug this x-value into f, and find that minimum value of y = f(x). :wink:

Eliz.
 
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