Match Each Transformation Of The Function \[$ F \$\] With A Feature Of The Transformed Function.1. Domain Of \[$ (5, \infty) \$\]2. Decreases As \[$ X \$\] Increases3. Range Of \[$ (5, \infty) \$\]4. \[$ X
Introduction
In mathematics, transformations of functions refer to the changes made to a function to create a new function. These changes can be in the form of shifts, stretches, compressions, or reflections. Understanding the transformations of functions is crucial in mathematics, as it helps in analyzing and solving problems related to functions. In this article, we will discuss the transformations of the function { f $}$ and match each transformation with a feature of the transformed function.
Domain of the Function
The domain of a function is the set of all possible input values for which the function is defined. In the case of the function { f $}$, the domain is given as { (5, \infty) $}$. This means that the function is defined for all values of { x $}$ greater than 5.
Decreases as { x $}$ Increases
A function that decreases as { x $}$ increases is a function that has a negative slope. In other words, as the input value { x $}$ increases, the output value of the function decreases. This is a characteristic of the function { f $}$, which decreases as { x $}$ increases.
Range of the Function
The range of a function is the set of all possible output values for which the function is defined. In the case of the function { f $}$, the range is given as { (5, \infty) $}$. This means that the function takes on all values greater than 5 as its output.
{ x $}$ Shifts
A { x $}$ shift refers to a transformation that moves the graph of a function to the left or right. In the case of the function { f $}$, the { x $}$ shift is given as 5. This means that the graph of the function is shifted 5 units to the right.
{ y $}$ Shifts
A { y $}$ shift refers to a transformation that moves the graph of a function up or down. In the case of the function { f $}$, there is no { y $}$ shift mentioned. This means that the graph of the function is not shifted up or down.
Stretches and Compressions
A stretch or compression refers to a transformation that changes the scale of a function. In the case of the function { f $}$, there is no mention of a stretch or compression. This means that the graph of the function is not stretched or compressed.
Reflections
A reflection refers to a transformation that flips a function over a line. In the case of the function { f $}$, there is no mention of a reflection. This means that the graph of the function is not reflected over a line.
Conclusion
In conclusion, the transformations of the function { f $}$ include a { x $}$ shift of 5, a domain of { (5, \infty) $}$, a range of { (5, \infty) $}$, and a decrease as { x $}$ increases. These transformations help in understanding the changes made to the original function and how they affect the graph of the function.
Matching the Transformations with Features
| Transformation | Feature |
|---|---|
| { x $}$ shift of 5 | Domain of { (5, \infty) $}$ |
| Domain of { (5, \infty) $}$ | { x $}$ shift of 5 |
| Range of { (5, \infty) $}$ | { x $}$ shift of 5 |
| Decreases as { x $}$ increases | { x $}$ shift of 5 |
Example Problems
- If the function { f $}$ has a domain of { (5, \infty) $}$ and a range of { (5, \infty) $}$, what is the { x $}$ shift of the function?
- If the function { f $}$ decreases as { x $}$ increases, what is the { x $}$ shift of the function?
- If the function { f $}$ has a { x $}$ shift of 5, what is the domain of the function?
Answer Key
- The { x $}$ shift of the function is 5.
- The { x $}$ shift of the function is 5.
- The domain of the function is { (5, \infty) $}$.
Final Thoughts
Frequently Asked Questions
Q1: What is a transformation of a function?
A1: A transformation of a function is a change made to a function to create a new function. These changes can be in the form of shifts, stretches, compressions, or reflections.
Q2: What are the different types of transformations?
A2: The different types of transformations include:
- Shifts: Moving the graph of a function to the left or right.
- Stretches and Compressions: Changing the scale of a function.
- Reflections: Flipping a function over a line.
- Rotations: Rotating a function around a point.
Q3: What is a { x $}$ shift?
A3: A { x $}$ shift refers to a transformation that moves the graph of a function to the left or right. For example, if a function has a { x $}$ shift of 2, the graph of the function will be shifted 2 units to the right.
Q4: What is a { y $}$ shift?
A4: A { y $}$ shift refers to a transformation that moves the graph of a function up or down. For example, if a function has a { y $}$ shift of 3, the graph of the function will be shifted 3 units up.
Q5: What is a stretch or compression?
A5: A stretch or compression refers to a transformation that changes the scale of a function. For example, if a function is stretched by a factor of 2, the graph of the function will be twice as wide as the original graph.
Q6: What is a reflection?
A6: A reflection refers to a transformation that flips a function over a line. For example, if a function is reflected over the x-axis, the graph of the function will be flipped upside down.
Q7: How do transformations affect the graph of a function?
A7: Transformations can change the shape, size, and position of the graph of a function. For example, a shift can move the graph of a function to a new location, while a stretch or compression can change the scale of the graph.
Q8: Can transformations be combined?
A8: Yes, transformations can be combined to create more complex transformations. For example, a function can be shifted and then stretched or compressed.
Q9: How do transformations affect the domain and range of a function?
A9: Transformations can change the domain and range of a function. For example, a shift can change the domain and range of a function, while a stretch or compression can change the scale of the domain and range.
Q10: Can transformations be used to solve problems?
A10: Yes, transformations can be used to solve problems. For example, a transformation can be used to find the inverse of a function or to solve a system of equations.
Example Problems
- If a function has a { x $}$ shift of 3, what is the new domain of the function?
- If a function is stretched by a factor of 2, what is the new scale of the function?
- If a function is reflected over the x-axis, what is the new graph of the function?
Answer Key
- The new domain of the function is { (3, \infty) $}$.
- The new scale of the function is twice as wide as the original graph.
- The new graph of the function is flipped upside down.
Final Thoughts
In conclusion, transformations of functions are an important concept in mathematics. Understanding the different types of transformations and how they affect the graph of a function is crucial in solving problems and analyzing functions.