Questions and Answers
All functions have a reciprocal that is also a function.
True
The reciprocal for a function is a reflection over the y and x axis.
False
If the original function has a root, then the reciprocal function will have a vertical asymptote.
True
Invariant points are used to graph reciprocal functions and happen when the point does not change after the transformation.
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If a function is a line, then its reciprocal is also a line.
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All functions have an inverse that is also a function.
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The inverse for a function is a reflection over the y and x axis.
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If the domain of a function is all real numbers except 5, then the range of its inverse is all real numbers except 5.
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True or false.
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True or false.
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True or false.
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True or false.
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True or false.
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True or false.
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True or false.
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True or false.
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When algebraically solving f(-x), what do you do?
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When algebraically solving -f(x), what do you do?
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When algebraically solving -f(-x), what do you do?
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When algebraically solving 1/f(x), what do you do?
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When algebraically solving f^-1(x), what do you do?
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When transforming a table of points for f(-x), what do you do?
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When transforming a table of points for -f(x), what do you do?
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When transforming a table of points for -f(-x), what do you do?
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When transforming a table of points for 1/f(x), what do you do?
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When transforming a table of points for f^-1(x), what do you do?
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When graphically transforming f(-x), what do you do?
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When graphically transforming -f(x), what do you do?
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When graphically transforming -f(-x), what do you do?
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When graphically transforming 1/f(x), what do you do?
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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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Description
Test your knowledge of transformation of functions in Honors Algebra 2 with these flashcards. This quiz includes true or false statements about properties and behaviors of reciprocal functions. Strengthen your understanding of function transformations and their graphical implications.
