Math Help - Range of a Function

1. Range of a Function

I pretty much can find the domain of certain basic functions. I understand the domain and range in the form of the point (x,y) = (domain, range). However, I just don't comprehend the idea of finding the range of any function.

Here are two questions:

(1) Find the range of f(x) = -(sqrt{-2x + 3})

(2) Find the range of y = -x^4 + 4

2. Originally Posted by magentarita
I pretty much can find the domain of certain basic functions. I understand the domain and range in the form of the point (x,y) = (domain, range). However, I just don't comprehend the idea of finding the range of any function.

Here are two questions:

(1) Find the range of f(x) = -(sqrt{-2x + 3})

(2) Find the range of y = -x^4 + 4
The best way of finding the range of a function is to draw the graph.

(1) (-oo, 0]

The graph is a sideways parabola. Note that:

* when x = 3/2, y = 0.
* As x --> -oo, y --> -oo.

(2) (-oo, 4]

The graph looks similar to an upside down parabola ......

3. yes but....

Originally Posted by mr fantastic
The best way of finding the range of a function is to draw the graph.

(1) (-oo, 0]

The graph is a sideways parabola. Note that:

* when x = 3/2, y = 0.
* As x --> -oo, y --> -oo.

(2) (-oo, 4]

The graph looks similar to an upside down parabola ......
Yes, but what is the basic idea in terms of the range of a function?

4. Originally Posted by magentarita
Yes, but what is the basic idea in terms of the range of a function?
The range is all the possible values of y. It's easiest to see them by drawing a graph.

5. but...

Originally Posted by mr fantastic
The range is all the possible values of y. It's easiest to see them by drawing a graph.
This is what all math books say but I would like the range to be broken down into words that I understand.

6. What do you understand for "all possible values for $y$?" For example, consider the function $f(x)=\frac1{x+1}.$ Note that $\not\exists\,x\in\mathbb R\mid f(x)=0,$ hence $y$ in this case, can't take the value zero.

Following up on mr fantastic's words, graphing, it's one of the best ways to find the range quickly.

7. Originally Posted by magentarita
(1) Find the range of f(x) = -(sqrt{-2x + 3})
The range is all $y$ such that the equation $y=-\sqrt{-2x+3}$ has a solution in the domain (i.e. $x\leq 1.5$). This gives $-y = \sqrt{-2x+3}$. In order for there to be a solution we need $-y\geq 0\implies y\leq 0$. Thus, $y^2 = -2x+3$. Which gives $x = \tfrac{1}{2}(3 - y^2)$ and this is of course in the domain. Thus, $y\leq 0$ is the range.

(2) Find the range of y = -x^4 + 4[
The range here is any real number. Thus, we are asking to find for what $y$ is the equation $y = -x^4 + 4$ solvable? This implies $x^4 = 4 - y$ and in order to have a solution we need $4 - y \geq 0 \implies y\leq 4$.

8. Originally Posted by magentarita
This is what all math books say but I would like the range to be broken down into words that I understand.
Dr. Math put it this way:

"Domain and range are just two different words for "how far something extends"; specifically, a king's domain is the territory he controls, and an animal's range is the region it wanders through. So it makes some sense that the set of numbers a function "controls" would be called its domain, and the set through which its value can wander is called its range."

Merriam-Webster puts it this way:

Domain: a territory over which dominion is exercised ; the set of lements to which a mathematical or logical variable is limited; specifically : the set on which a function is defined. (The word comes from the Latin word for "lordship".)

Range: a place that may be ranged over; an open region over which animals (as livestock) may roam and feed; the region throughout which a kind of organism or ecological community naturally lives or occurs; the set of values a function may take on; the class of admissible values of a variable.

In mathematics:

The domain of a function f(x) is usually fairly easy to find. It is the set of all the numbers x that can be put into f(x) and have the result make sense. That is, the x values that don't make some expression inside a square root sign negative, or that don't make a denominator zero, and so on.

The range is the set of all values f(x) can take, as x takes every value in the domain.

Example: $y=x^2-2$

As for the domain, there are no restrictions. You can assign any real number to x. So the domain is "all real numbers"

The range is the set of all values y can take, as x takes every value in the domain. In this problem, you know that the square of a number is greater than or equal to 0. Could y take the value -3?

If we try to solve

$-3 = x^2 - 2$

we get

$-1 = x^2$

which is impossible to solve, so -3 is not in the range. One way to find the values of y that are possible is to try solving the equation for x:

$y = x^2 - 2$

$y + 2 = x^2$

$x = \pm \sqrt{y+2}$

Now you can use the logic you used for the domain: what values of y will let this formula make sense? The radicand must be greater than or equal to 0.

$y+2 \ge 0$
$y \ge -2$

Range = $\{y|y \ge -2\} \ \ or \ \ [-2, +\infty)$

9. much better............

Originally Posted by Krizalid
What do you understand for "all possible values for $y$?" For example, consider the function $f(x)=\frac1{x+1}.$ Note that $\not\exists\,x\in\mathbb R\mid f(x)=0,$ hence $y$ in this case, can't take the value zero.

Following up on mr fantastic's words, graphing, it's one of the best ways to find the range quickly.

Thanks for breaking this up for me.