a.Evaluate 2f(3)+3f(-3), b. What is the domain of f?, c. What is the smallest possible value of f(x)?, d. What is the range of f?

Question

Consider the function f(x)=(2x*x)- 4x+9.

Answer 1

As far as your question goes.

 

a. To evaluate 2f(3) + 3f(-3), there are two parts.

2f(3) means to plug the value of 3 into the function a then multiply by 2.

3f(-2) means to plug the value of -3 into the function and then multiply by 3.

 

Particularly note that -3 squared is 9 and not -9.

 

b. Domain

Looking at your function f(x) = (2x²) -4x+9 there are no restrictions to your domain.

Restrictions occur in two cases...when you divide by zero, or when you have a value under a even based radical like a square root.

 

Since there are a no restrictions, you will have a domain of all real numbers.

 

3) What is the smallest value?

 

Your function follows the general form of a parabola ax²+bx+c.  It is a quadratic function as can been seen from highest exponent, the term x².

 

For a parabola, your highest or lowest value for f(x) will occur at the vertex.  The vertex formula is (-b/2a , f(-b/2a))

 

What does this mean?

Figure out -b/2a

Then plug that number into the function to find the value for f(x)

 

For your problem, a is 2,  b is 4.

-b/2a = -4/2(2)  = -1

 

plug -1 into your function

f(-1) = (2 * -1 * -1) - 4 (-1) +9

 

=2+4+9 = 15

 

So,  f(x) = 15.

 

Note: A common mistake occurs from not putting all the values in or dropping your signs.

 

 

d) Now that you know your smallest possible value you can find your range (all possible values for f(x).

 

Since 15 is the smallest, remembering that the parabola makes a U shape, the ends of the parabola go on forever.

So your range is the y values from 15 to infinity.

 

We write this [15, ∞)

 

Hope that helps!

 

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