Might be easier than I'm making it but still need help.

[bd40]

This is the problem. Suppose f(x)= -x^2+1 and the domain of the function is the real numbers. Give an example of a number that is NOT in the range of f.

First I got 1-x^2

Then I did it again and I got x-1=x

Can anyone tell me what the answer would be? I was afraid to try again because I might end up with 0 or some other crazy number.

 

[HallsofIvy]

Do you understand what the question is? You are asked to "give a number that is NOT in the range" so I presume you understand that your answer must be a number. Do you understand what the "range" of a function is? The range is the set of all function values of the function- if you write y= f(x) then the "range" is the set of all possible values of y.

"First I got 1-x^2
Then I did it again and I got x-1=x"
What did you "do again"?

If you were to graph the function, what would its graph look like?

You know, I suppose, that x^2 is never negative. That means that y= 1- x^2 is always 1 minus a non-negative number so y is always less than what number?

 

[bd40]

Do you understand what the question is? You are asked to "give a number that is NOT in the range" so I presume you understand that your answer must be a number. Do you understand what the "range" of a function is? The range is the set of all function values of the function- if you write y= f(x) then the "range" is the set of all possible values of y.

"First I got 1-x^2
Then I did it again and I got x-1=x"
What did you "do again"?

If you were to graph the function, what would its graph look like?

You know, I suppose, that x^2 is never negative. That means that y= 1- x^2 is always 1 minus a non-negative number so y is always less than what number?

I thought the domain was all the x values and the y was the range values but I still don't understand. Does this mean that 1 is the lowest number for f(x) or could it also be a fraction as long as it isn't a negative? Therefore y=.5^2+1 giving me the answer of y=1.25 ?

 

[JeffM]

I thought the domain was all the x values and the y was the range values but I still don't understand. Does this mean that 1 is the lowest number for f(x) or could it also be a fraction as long as it isn't a negative? Therefore y=.5^2+1 giving me the answer of y=1.25 ?

This is a problem where things are so simple that you miss the answer through overcomplicating things.

Let's review the basics. Consider the function y = f(x). It is merely a RULE for associating a value inside the parentheses (the argument or input) with a resulting or output value.

f(x) = x + 2 MEANS associate whatever x is with 2 more than x. So x is the input, and x + 2 is the output.

f(x) = x MEANS associate whatever x is with itself. So x is the input and also the output.

The domain of the function are the values of the input x to which the rule applies. They are permissable values for x.

The range of the function are the values that the rule may generate as output. They are possible values for y.

So let's think a moment about the function \(\displaystyle y = f(x) = 1 - x^2\).

Are there any numbers to which the rule cannot be applied?

No, the rule will work for any number so we say the domain of that function is all real numbers. (We could if we wanted restrict it some part of the real numbers, but this problem says not to do that.)

As Halls pointed out, x² is non-negative for every value x. So - x² is non-positive for every value of x.

Now let's think about 1 - x².

\(\displaystyle x^2 \ge 1 \implies - x^2 \le - 1 \implies f(x) = 1 - x^2 \le 1 - 1 = 0.\) With me so far?

Of course x² may be less than 1 but it is not less than 0.

\(\displaystyle 0 \le x^2 < 1 \implies -1 < - x^2 \le 0 \implies 1 - 1 < 1 - x^2 \le 1 + 0 \implies 0 < f(x) \le 1.\)

Now we can specify the range of f(x). Can y be negative? Can y equal zero? Can y be positive? Is there any number that y cannot exceed?

So the range is what?

Consequently, there are an infinite number of numbers that are outside the range of the function. The problem asks you to name one of those infinite number of numbers. So tell me what one of them is.

 

[HallsofIvy]

I thought the domain was all the x values and the y was the range values but I still don't understand. Does this mean that 1 is the lowest number for f(x) or could it also be a fraction as long as it isn't a negative? Therefore y=.5^2+1 giving me the answer of y=1.25 ?

The function you gave originally was \(\displaystyle y= -x^2+ 1\), NOT \(\displaystyle y= x^2+ 1\)! There's a big difference! If x= 0 then both give y= 1 but with y= -x^2+ 1, if x is anything other than 0, \(\displaystyle x^2\) is positive so \(\displaystyle -x^2+ 1\) is 1 minus something. 1 is the largest value of y.