Range of Rational Functions....3

In summary, the function y = sqrt{2x - 4} can be graphed, but the range of the function is not clear. To find the inverse of this function, x^2 + 4 must be restricted to be within the domain of the inverse function, y = 2x - 4.
  • #1
mathdad
1,283
1
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.
 
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  • #2
RTCNTC said:
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 ...

5014985_orig.jpg


... and how these functions are transformed.

Transformations-of-Functions
 
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  • #3
I will dig deeper into this topic when I get to rational functions and graphs.
 
  • #4
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?
 
  • #5
RTCNTC said:
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)
 
  • #6
Since the domain of the inverse is x > or = 0, then the range of the original function is the same answer.
 
  • #7
RTCNTC said:
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.
 
  • #8
MarkFL said:
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?
 
  • #9
RTCNTC said:
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.
 
  • #10
Thank you everyone. Good information here.
 

Related to Range of Rational Functions....3

1. What is the definition of a rational function?

A rational function is a mathematical function in the form of f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions. This means that the numerator and denominator of the function are both made up of terms with variables raised to non-negative integer powers.

2. How do you find the domain of a rational function?

The domain of a rational function is all the values of x for which the function is defined. To find the domain, set the denominator of the function equal to zero and solve for x. The domain will be all real numbers except for the values that make the denominator equal to zero.

3. What is the significance of the vertical asymptotes in a rational function?

Vertical asymptotes are important in rational functions because they represent values of x that make the denominator of the function equal to zero. These values are not included in the domain of the function and can cause the function to have vertical gaps or jumps on the graph.

4. How do you determine the horizontal asymptote of a rational function?

The horizontal asymptote of a rational function is determined by looking at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be y = 0. If the degrees are equal, the horizontal asymptote will be the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there will be no horizontal asymptote.

5. What is the process for graphing a rational function?

To graph a rational function, start by finding the domain, vertical asymptotes, and horizontal asymptote. Then, choose a few values for x and plug them into the function to find corresponding y values. Plot these points and use them to sketch the graph, making sure to include any vertical asymptotes. Lastly, check for any other important features such as x-intercepts or holes in the graph.

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