MHB Range of Rational Functions....3

[mathdad]

Find the range of y = sqrt{2x - 4}.

I need the steps. According to the textbook, graphing the function leads to finding the range. This may be true for others but not for me. I am not clear on the range idea.

 

[skeeter]

Find the range of y = sqrt{2x - 4}.

I need the steps. According to the textbook, graphing the function leads to finding the range. This may be true for others but not for me. I am not clear on the range idea.

parent function is $y = \sqrt{x}$ ... you should be able to sketch this function by hand

$y = \sqrt{2x-4} = \sqrt{2(x-2)} = \sqrt{2} \cdot \sqrt{x-2}$

the parent function is horizontally shifted right 2 units and vertically stretched by a factor of $\sqrt{2}$

The domain requires $x-2 \ge 0$.

The range will be $y \ge 0$.

recommend you learn the graphs of parent functions ...

... and how these functions are transformed.

Transformations-of-Functions

 

[mathdad]

I will dig deeper into this topic when I get to rational functions and graphs.

 

[mathdad]

Can I find the inverse of the given function, find the domain of the inverse which is the range of the original function?

Let me see.

y = sqrt{2x - 4}

x = sqrt{2y - 4}

x^2 = [sqrt{2y - 4}]^2

x^2 = 2y - 4

x^2 + 4 = 2y

(x^2 + 4)/2 = y

1. Is the inverse of the original function y = (x^2 + 4)/2?

2. What is the domain of the inverse function, which is the range of the original function?

 

[skeeter]

Can I find the inverse of the given function, find the domain of the inverse which is the range of the original function?

1. Is the inverse of the original function y = (x^2 + 4)/2?

2. What is the domain of the inverse function, which is the range of the original function?

1. it is for $x \ge 0$

2. domain stated in part (1)

 

[mathdad]

Since the domain of the inverse is x > or = 0, then the range of the original function is the same answer.

 

[MarkFL]

Since the domain of the inverse is x > or = 0, then the range of the original function is the same answer.

This is what I was hinting at earlier regarding functions that aren't one-to one. We have to restrict the domain for such functions to get a valid inverse.

 

[mathdad]

This is what I was hinting at earlier regarding functions that aren't one-to one. We have to restrict the domain for such functions to get a valid inverse.

Can you provide a list of functions where restricting the domain is needed?

 

[skeeter]

Can you provide a list of functions where restricting the domain is needed?

Any function that is not one-to-one ... a "list" of functions with that property would be rather extensive and not all inclusive.

 

[mathdad]

Thank you everyone. Good information here.