Examples of additive inverse in the following topics:
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- In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices.
- Otherwise, it is equal to the additive inverse of its minor: $C_{ij}=-M_{ij}$
- Since $i+j=5 $ is an odd number, the cofactor is the additive inverse of its minor: $-(13)=-13$
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- Functions that have an additive inverse can be classified as odd or even depending on their symmetry properties.
- For a function to be classified as one or the other, it must have an additive inverse.
- In addition, for every point $(x,y)$ on the graph, the corresponding point $(-x,-y)$ is also on the graph.
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- Recognize whether a function has an inverse by using the horizontal line test
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- Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
- Below is the graph of the parabola and its "inverse."
- Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.
- Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
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- To find the inverse function, switch the $x$ and $y$ values, and then solve for $y$.
- An inverse function, which is notated $f^{-1}(x)
$, is defined as the inverse function of $f(x)$ if it consistently reverses the $f(x)$ process.
- In general, given a function, how do you find its inverse function?
- Remember that an inverse function reverses the inputs and outputs.
- A function's inverse may not always be a function.
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- Each trigonometric function has an inverse function that can be graphed.
- To use inverse trigonometric functions, we need to understand
that an inverse trigonometric function “undoes” what the original
trigonometric function “does,” as is the case with any other function
and its inverse.
- The inverse of sine is arcsine (denoted $\arcsin$), the inverse of cosine is arccosine (denoted $\arccos$), and the inverse of tangent is arctangent (denoted $\arctan$).
- An exponent of $-1$ is used to indicate an inverse function.
- The inverse sine function can also be written $\arcsin x$.
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- The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.
- In order to solve for the missing acute angle, use the same three trigonometric functions, but, use the inverse key ($^{-1}$on the calculator) to solve for the angle ($A$) when given two sides.
- (Soh from SohCahToa) Write the equation and solve using the inverse key for sine.
- Recognize the role of inverse trigonometric functions in solving problems about right triangles
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- Two variables in direct variation have a linear relationship, while variables in inverse variation do not.
- Inverse variation is
the opposite of direct variation.
- As an example, the time taken for a journey is inversely proportional to the speed of travel.
- Thus, an inverse relationship cannot be represented by a line with constant slope.
- Relate the concept of slope to the concepts of direct and inverse variation
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- Note also that only square matrices can have an inverse.
- Example: Find the inverse of: $\begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix}$
- It establishes $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ as the inverse that is being looked for, by asserting that it fills the definition of an inverse matrix.
- In some cases, the inverse of a square matrix does not exist.
- Practice finding the inverse of a matrix and describe its properties
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- A system of equations can be readily solved using the concept of the inverse matrix and matrix multiplication.
- A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication.
- Now, in order to determine the values of $x$, $y$, and $z$, we simply multiply the inverse of $[A]$ times $[B]$.
- Then calculate $[A^{-1}][B]$, that is, the inverse of matrix $[A]$, multiplied by matrix $[B]$.
- Practice using inverse matrices to solve a system of linear equations