Range of Rational Functions....1

In summary, to find the range of a function, you can shift it left/right, but it will not affect its range. The inverse function will have the same range.
  • #1
mathdad
1,283
1
Finding the range of functions can be complicated. Rational functions can be complicated. I have a terribly hard time finding the range of functions.

Find the range of y = 1/(x + 4).

How is this done?

I want the steps.
 
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  • #2
Shifting a function left/right won't affect its range...the given function will then have the same range as:

\(\displaystyle y=\frac{1}{x}\)

If you are familiar with the graph of this function, its range is readily apparent. If not, sometimes we can find the inverse function, whose domain is the range of the original. The function above is special in that it is its own inverse. So, take the domain of the above function, and you have its range as well, which will be the range of the original function you gave. :D
 
  • #3
y = 1/(x + 4)

Set x + 4 = 0.

x + 4 - 4 = -4

x = -4

The domain is ALL REALL NUMBERS except that x cannot be -4.

If I understood correctly, the range is ALL REAL NUMBERS except for -4.

Is this right?
 
  • #4
RTCNTC said:
y = 1/(x + 4)

Set x + 4 = 0.

x + 4 - 4 = -4

x = -4

The domain is ALL REALL NUMBERS except that x cannot be -4.

If I understood correctly, the range is ALL REAL NUMBERS except for -4.

Is this right?

I was talking about the following function:

\(\displaystyle y=\frac{1}{x}\)

being its own inverse. :D
 
  • #5
For y = 1/x, the domain is ALL REALL NUMBERS except for 0.

So, the range of y = 1/( x + 4) is ALL REAL NUMBERS except for 0.

Yes? No?
 
  • #6
Are you saying that I need to find the inverse of
y = 1/(x + 4)?

Let me see.

x = 1/(y + 4)

Then y + 4 = 0.

y + 4 - 4 = -4

y = -4

The domain of the inverse function is ALL REAL NUMBERS except for -4.

This is the range of the given function.

Correct?
 
  • #7
RTCNTC said:
For y = 1/x, the domain is ALL REALL NUMBERS except for 0.

So, the range of y = 1/( x + 4) is ALL REAL NUMBERS except for 0.

Yes? No?

Yes, that's right.

RTCNTC said:
Are you saying that I need to find the inverse of
y = 1/(x + 4)?

I'm saying finding the inverse and taking its domain as the range of the original is one way to go about it. :D

RTCNTC said:
Let me see.

x = 1/(y + 4)

Now you want to solve for $y$, and then you have the inverse function.
 
  • #8
x = 1/(y + 4)

x(y + 4) = 1

xy + 4y = 1

y(x + 4) = 1

y = 1/(x + 4)

Like you said, it is its own inverse.

The domain of y = 1/(x + 4) is ALL REAL NUMBERS except for -4.
 
  • #9
RTCNTC said:
x = 1/(y + 4)

x(y + 4) = 1

xy + 4y = 1

y(x + 4) = 1

y = 1/(x + 4)

Like you said, it is its own inverse.

The domain of y = 1/(x + 4) is ALL REAL NUMBERS except for -4.

Let's go back to:

\(\displaystyle x=\frac{1}{y+4}\)

This means:

\(\displaystyle y+4=\frac{1}{x}\)

Or:

\(\displaystyle y=\frac{1}{x}-4\)

This demonstrates that the shift left/right has no bearing on the range.

Can you spot your algebraic error when solving for $y$ to get the inverse?
 
  • #10
Let me start again.

y = 1/(x + 4)

x = 1/(y + 4)

x(y + 4) = 1

xy + 4x = 1

xy = 1 - 4x

y = (1 - 4x)/x

The domain of the inverse is ALL REAL NUMBERS except for 0.

This is the range of the original function.
 

Related to Range of Rational Functions....1

1. What is a rational function?

A rational function is a mathematical expression that is written as the ratio of two polynomial functions. In other words, it is a fraction where the numerator and denominator are both polynomial functions.

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

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

3. What is the range of a rational function?

The range of a rational function is the set of all possible output values of the function. In other words, it is the set of all real numbers that the function can take on as its output. The range of a rational function can be determined by examining the behavior of the function as the input approaches positive and negative infinity.

4. How do you simplify a rational function?

To simplify a rational function, factor both the numerator and denominator and cancel out any common factors. This will result in a simplified form of the rational function, which may be easier to work with or graph.

5. What is the difference between a rational function and an irrational function?

A rational function is a function that can be written as the ratio of two polynomial functions, while an irrational function is a function that cannot be written in this form. Irrational functions often involve roots or exponents, and their graphs may not have a defined shape or pattern.

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