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Questions and Answers
All functions have a reciprocal that is also a function.
All functions have a reciprocal that is also a function.
True (A)
The reciprocal for a function is a reflection over the y and x axis.
The reciprocal for a function is a reflection over the y and x axis.
False (B)
If the original function has a root, then the reciprocal function will have a vertical asymptote.
If the original function has a root, then the reciprocal function will have a vertical asymptote.
True (A)
Invariant points are used to graph reciprocal functions and happen when the point does not change after the transformation.
Invariant points are used to graph reciprocal functions and happen when the point does not change after the transformation.
If a function is a line, then its reciprocal is also a line.
If a function is a line, then its reciprocal is also a line.
All functions have an inverse that is also a function.
All functions have an inverse that is also a function.
The inverse for a function is a reflection over the y and x axis.
The inverse for a function is a reflection over the y and x axis.
If the domain of a function is all real numbers except 5, then the range of its inverse is all real numbers except 5.
If the domain of a function is all real numbers except 5, then the range of its inverse is all real numbers except 5.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
True or false.
When algebraically solving f(-x), what do you do?
When algebraically solving f(-x), what do you do?
When algebraically solving -f(x), what do you do?
When algebraically solving -f(x), what do you do?
When algebraically solving -f(-x), what do you do?
When algebraically solving -f(-x), what do you do?
When algebraically solving 1/f(x), what do you do?
When algebraically solving 1/f(x), what do you do?
When algebraically solving f^-1(x), what do you do?
When algebraically solving f^-1(x), what do you do?
When transforming a table of points for f(-x), what do you do?
When transforming a table of points for f(-x), what do you do?
When transforming a table of points for -f(x), what do you do?
When transforming a table of points for -f(x), what do you do?
When transforming a table of points for -f(-x), what do you do?
When transforming a table of points for -f(-x), what do you do?
When transforming a table of points for 1/f(x), what do you do?
When transforming a table of points for 1/f(x), what do you do?
When transforming a table of points for f^-1(x), what do you do?
When transforming a table of points for f^-1(x), what do you do?
When graphically transforming f(-x), what do you do?
When graphically transforming f(-x), what do you do?
When graphically transforming -f(x), what do you do?
When graphically transforming -f(x), what do you do?
When graphically transforming -f(-x), what do you do?
When graphically transforming -f(-x), what do you do?
When graphically transforming 1/f(x), what do you do?
When graphically transforming 1/f(x), what do you do?
When graphically transforming f^-1(x), what do you do?
When graphically transforming f^-1(x), what do you do?
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Study Notes
Transformation of Functions
- All functions possess a reciprocal function, which is itself also a function.
- The assertion that the reciprocal is a reflection over both the y and x axes is incorrect.
- If an original function has a root, its reciprocal function will have a vertical asymptote.
- Invariant points, where points remain unchanged during transformation, are crucial for graphing reciprocal functions.
- Reciprocal functions differ in nature: a linear function’s reciprocal is not necessarily linear; similar holds for quadratics and rational functions.
- Not all functions have an inverse that is also a function.
- The notion that the inverse is a reflection over the y and x axes is false.
- A function's inverse retains the domain and range relationship—in this case, if the domain excludes a number, so will the range of its inverse.
- The vertical line test is not a valid method to determine if a function has an inverse that is itself a function.
- A function's inverse also does not follow the same type (linear, quadratic) as the original function.
- Standard processes for transformations include algebraic methods of substituting and adjusting function expressions.
Algebraic Transformation Techniques
- To solve for f(-x), substitute -x into the function.
- To solve for -f(x), apply the negative to the entire function expression.
- For -f(-x), first substitute -x and then negate the entire expression.
- When solving 1/f(x), take the reciprocal of the entire function's expression.
- To find the inverse f^-1(x), switch the variables x and y and resolve for y.
Point Transformation for Functions
- Transforming f(-x) alters the point coordinates to (-x, y).
- Transforming -f(x) changes the points to (x, -y).
- For -f(-x), points are transformed to (-x, -y).
- The process for 1/f(x) involves taking the reciprocal of the y-values and determining the vertical asymptote when transforming point tables.
- When deriving f^-1(x), swap the x and y values of the points.
Graphical Transformation Techniques
- Graphically transforming f(-x) requires reflecting the function over the y-axis.
- Reflecting the function -f(x) takes place over the x-axis.
- For -f(-x), the reflection occurs over both the x and y axes.
- Transforming 1/f(x) involves a series of changes: x-intercepts become asymptotes, smaller y-values switch with larger ones, and local maxima become minima (and vice versa).
- For graphing f^-1(x), perform reflections over the line y=x, and swap intercepts accordingly while leaving points on the line y=x unchanged.
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