Transformation of Functions: How Do Domain and Range Change?

In summary: You can see that they have the same shape, but the red one has been shifted to the left by 3 units. The domain of the blue function is [0, 2], and the domain of the red function is [-3, -1].
  • #1
DumpmeAdrenaline
78
2
I want to understand how the domain and range change upon applying transformations like (left/right shifts, up/down shifts, and vertical/horizontal stretching/compression) on functions.
Let f(x)=2-x if 0 ≤x ≤2 and 0 otherwise.
I want to describe the following functions 1) f(-x) 2) -f(x) 3) f(x+3) 4) f(x)+3 5) f(2x) 6) 2f(x) 7) f(2x+3)
The rule generates an output by multiplying the input by -1 and adding the result to 2.
1) f operates on the closed interval [0,2] as its domain. With f(-x) the domain then changes to [-2,0] so that upon multiplying each real number in this interval by -1 we obtain the same domain [0,2] and the same image [0,2] as f(x). f(-x)=2-(-x)=2+x.
2) -f(x) change the sign of the output keeping the input.
3) f(x+3) Each input in the closed interval [0,2] moves by 3 units, such that the domain changes to 3 ≤x+3 ≤5. However, under this domain the function generates an output of 0. As a result the domain of f(x+3) becomes [0,-1] so that upon adding 3 units to we wind up with the same domain and range as f(x).
4) Move every point up by 3 units
5) The domain of f(2x) shrinks to 0<x<1 so that when multiply each real number we obtain the same domain and range as f(x). f(2x)=2-2x
6) Multiply each output by 2 keeping the input.
7) Composed transformation of shrinking by 1/2 followed by a shift 3 units to the left resulting in a domain of [-3,-1].
In all of the above are we changing the rule f?
 
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  • #2
DumpmeAdrenaline said:
I want to describe
You also want to learn how to post math using a little ##\ \LaTeX\ ##.
See tutorial. It's really easy, and it's fun. Enclose in ## for in-line math and in $$ for displayed math. Example:
Code:
$$1) \quad  f(-x) \\ 2) \quad -f(x) \\ 3) \quad f(x+3) \\
4) \quad f(x)+3 \\5) \quad f(2x) \\6) \quad 2f(x) \\7) \quad f(2x+3)$$
yields
$$1) \quad f(-x) \\ 2) \quad -f(x) \\ 3) \quad f(x+3) \\4) \quad f(x)+3 \\5) \quad f(2x) \\6) \quad 2f(x) \\7) \quad f(2x+3)$$

[edit] this looks like ... because MathJax doesn't acknowledge the line feeds (\\\) and I don't understand what is screwing things up. :cry:

A better way to do this is with \begin{align} and let ##\TeX## do the numbering:
Code:
$$\begin{align}
 & f(-x)  \\  -&f(x) \\ &f(x+3) \\ &f(x)+3 \\&f(2x) \\2&f(x) \\ &f(2x+3)
\end{align}$$
$$\begin{align}
& f(-x) \\ -&f(x) \\ &f(x+3) \\ &f(x)+3 \\&f(2x) \\2&f(x) \\ &f(2x+3)
\end{align}$$

##\ ##
 
Last edited:
  • #3
DumpmeAdrenaline said:
3) f(x+3) Each input in the closed interval [0,2] moves by 3 units, such that the domain changes to 3 ≤x+3 ≤5.
No. Make a sketch to see. You doubled up: domain is ##0 \le x+3 \le 2## .

DumpmeAdrenaline said:
In all of the above are we changing the rule f?
No, we are not.

##\ ##
 
  • #4
BvU said:
No. Make a sketch to see. You doubled up: domain is
I reached the same answer but for the wrong reason.I thought that the x in f(x) is the same as x+3. Hence, why I added 3 to both sides of the inequality. So, we regard x+3 and x as just labels to the input?
 
  • #5
From post #1:
DumpmeAdrenaline said:
3) f(x+3) Each input in the closed interval [0,2] moves by 3 units, such that the domain changes to 3 ≤x+3 ≤5. However, under this domain the function generates an output of 0. As a result the domain of f(x+3) becomes [0,-1] so that upon adding 3 units to we wind up with the same domain and range as f(x).

BvU said:
No. Make a sketch to see. You doubled up: domain is ##0 \le x+3 \le 2## .
And further, ##0 \le x+3 \le 2 \Rightarrow -3 \le x \le -1##, so the domain for y = f(x + 3), is the translation left by 3 units of the domain for y = f(x).
DumpmeAdrenaline said:
I reached the same answer but for the wrong reason.I thought that the x in f(x) is the same as x+3.
You're dealing with two different functions, each of which is a translation by 3 units of the other.

Here's a sketch of y = f(x) (in blue) and y = f(x + 3) (in red).
graph.png
 

Related to Transformation of Functions: How Do Domain and Range Change?

What is the transformation of functions?

The transformation of functions refers to the process of manipulating a function by changing its shape, position, or orientation on a graph. This can be done by applying mathematical operations such as shifting, stretching, compressing, or reflecting the function.

What are the common types of transformations in functions?

The common types of transformations in functions include translation, dilation, reflection, and rotation. Translation involves shifting the function horizontally or vertically, while dilation involves stretching or compressing the function. Reflection is the process of flipping the function across a line, and rotation involves rotating the function around a fixed point.

How do you determine the effects of a transformation on a function?

The effects of a transformation on a function can be determined by understanding the rules for each type of transformation. For example, a horizontal translation of a function by a value of c will shift the function c units to the left or right, depending on the sign of c. Similarly, a vertical translation of a function by a value of d will shift the function d units up or down, depending on the sign of d.

What is the purpose of transforming functions?

The purpose of transforming functions is to visualize and analyze how changes to the input or output of a function affect its graph. This can help in understanding the behavior of a function and making predictions about its values. Transformations are also useful in creating new functions from existing ones, which can be used to model real-world situations.

How do transformations affect the domain and range of a function?

Transformations can affect the domain and range of a function by changing the values that the function can take on. For example, a horizontal translation can shift the domain of a function, while a vertical translation can shift the range. Dilation and reflection can also change the domain and range of a function. It is important to consider these effects when working with transformed functions.

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