Write a reciprocal function translated to the right by 3, has a vertical stretch of 2, and is reflected over the y-axis. Determine the domain and range.
consider f(x) = 1/x
domain: x≠0
range y≠0
translate right 3: 1/(x-3)
stretch: 2/(x-3)
reflect: 2/((-x)-3) = -2/(x+3)
domain: x+3≠0, so x ≠ -3
range: y≠0
see the graphs at
www.wolframalpha.com/input/?i=plot+y%3D1%2Fx%2C+y%3D2%2F%28-3-x%29
Why did the reciprocal function go to a party?
Because it heard there would be lots of reflections and translations!
Now, let's get into the "mathy" stuff. The reciprocal function is given by f(x) = 1/x. To translate it to the right by 3, we need to replace x with (x - 3). So the translated function becomes f(x) = 1/(x - 3).
To have a vertical stretch of 2, we multiply the function by 2. Hence, the new function is f(x) = 2/(x - 3).
Next, we need to reflect the function over the y-axis. To do this, we replace x with -x. The final function becomes f(x) = 2/(-x - 3).
Now, let's talk domain and range. The domain is all the values of x for which the function is defined. In this case, the function is defined for all real numbers except x = -3 (the value that makes the denominator zero).
So, the domain is the set of all real numbers except -3, written as (-∞, -3) U (-3, ∞).
The range, on the other hand, is all the possible values that the function can output. Since the function is reflected over the y-axis, the sign of the output values is flipped. Therefore, the range is all real numbers except 0.
So, the range is the set of all real numbers except 0, written as (-∞, 0) U (0, ∞).
I hope that answers your question with a twist of humor!
To translate a reciprocal function to the right by 3, we subtract 3 from the independent variable (x). To reflect the function over the y-axis, we negate the reciprocal function itself (-f(x)). And to apply a vertical stretch of 2, we multiply the entire function by 2.
The general form of a reciprocal function is:
f(x) = a/(x - h) + k
Applying the given transformations, the function becomes:
f(x) = 2/(-x - 3)
The domain of a reciprocal function is all real numbers except the value(s) that make the denominator equal to zero. In this case, the denominator cannot be zero if -x - 3 ≠ 0. Solving for x:
-x - 3 ≠ 0
-x ≠ 3
x ≠ -3
So, the domain of the function is all real numbers except x = -3.
The range of a reciprocal function is all real numbers except the value(s) that make the function undefined. As the vertical stretch does not affect the range, the range of this function is all real numbers except y = 0.
Therefore, the domain of the function is all real numbers except x = -3, and the range is all real numbers except y = 0.
To find the equation of the reciprocal function with the given transformations, we can start with the general form of a reciprocal function:
f(x) = a / (x - h) + k
where a is the vertical stretch, (h, k) is the translation, and the reflection over the y-axis is represented by a negative value of a.
In this case, the translation to the right by 3 means h = -3, the vertical stretch of 2 means a = 2, and the reflection over the y-axis means a is negative.
Therefore, the equation of the reciprocal function is:
f(x) = -2 / (x + 3)
Now, let's determine the domain and range of the function.
The domain of a reciprocal function is all real numbers except for the values of x that make the denominator (x + h) equal to zero. In this case, x + 3 = 0, so x = -3. Therefore, the domain of the function is all real numbers except x = -3.
The range of a reciprocal function is all real numbers except for zero. Since the numerator is -2, the function will never be equal to zero. Therefore, the range of the function is all real numbers except zero.
In summary, the equation of the reciprocal function, after being translated to the right by 3, having a vertical stretch of 2, and being reflected over the y-axis is:
f(x) = -2 / (x + 3)
The domain is all real numbers except x = -3, and the range is all real numbers except zero.