Functions as Graphs - Algebra 2

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

It follows that is an odd function.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

It follows that is an odd function.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

;

the function cannot be even. This does allow for the function to be odd. However, if is odd, then, by definition,

, or

and is equal to its own opposite - the only such number is 0, so

.

This is not the case - - so the function is not odd either.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

;

the function cannot be even. This does allow for the function to be odd. However, if is odd, then, by definition,

, or

and is equal to its own opposite - the only such number is 0, so

.

This is not the case - - so the function is not odd either.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

;

the function cannot be even. This does allow for the function to be odd. However, if is odd, then, by definition,

, or

and is equal to its own opposite - the only such number is 0, so

.

This is not the case - - so the function is not odd either.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

;

the function cannot be even. This does allow for the function to be odd. However, if is odd, then, by definition,

, or

and is equal to its own opposite - the only such number is 0, so

.

This is not the case - - so the function is not odd either.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

It follows that is an odd function.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

is a rational function in simplest form whose denominator has a polynomial with degree greater than that of the polynomial in its numerator (2 and 1, respectively). The graph of such a function has as its horizontal asymptote the line of the equation .

To find the slope of an equation first get the equation in slope intercept form.

where,

represents the slope.

Thus

When determining how a the graph of a function will be translated, we know that anything that happens to x in the function will impact the graph horizontally, opposite of what is expressed in the function, whereas anything that is outside the function will impact the graph vertically the same as it is in the function notation.

For this graph:

The graph will move 3 spaces left, because that is the opposite sign of the what is connected to x directly.

Also, the graph will move down 2 spaces, because that is what is outside the function and the 2 is negative.

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Also, it is seen to be symmetrical about the -axis. Consequently, for each in the domain, - the function is even.

A function may only have one y-value for each x-value.

The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.

The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or ) value for each value of . The vertical line test determines how many (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function:

Example of an equation that is not a function: