Answer: For an even function even function A function is an even function if f of x is equal to f of −x for all the values of x. This means that the function is the same for the positive x-axis and the negative x-axis, or graphically, symmetric about the y-axis. https://www.cuemath.com › calculus › even-function

Contents

- 1 What makes a function even?
- 2 What is an example of an even function?
- 3 What is an odd function example?
- 4 Is a linear function even or odd?
- 5 What types of functions are even or odd?
- 6 Which function is an odd function?
- 7 What is an example of a function that isn’t even or odd?
- 8 Is 0 neither odd nor even?
- 9 Is 2.5 odd or even?
- 10 How do you identify odd numbers?
- 11 Is 67 odd or even?

### Even Function – Cuemath

, f(-x) = f(x), for all x, for an odd function odd function Odd Function. The odd functions are functions that return their negative inverse when x is replaced with –x. This means that f(x) is an odd function when f(-x) = -f(x). Some examples of odd functions are trigonometric sine function, tangent function, cosecant function, etc. https://www.cuemath.com › calculus › odd-functions

### Odd Functions – Cuemath

f(-x) = -f(x), for all x. If f(x) ≠ f(−x) and −f(x) ≠ f(−x) for some values of x, then f is neither even nor odd.

A function f is even if f(−x)=f(x), for all x in the domain of f. A function f is odd if f(−x)=−f(x), for all x in the domain of f.

## What makes a function even?

A function is called an even function if its graph is unchanged under reflection in the y-axis. Suppose f(x) is a function such that it is said to be an even function if f(-x) is equal to f(x).

## What is an example of an even function?

Odd Function – A function \(f\) is called an odd function if \begin f(x)=-f(-x) \end for all \(x \) in the domain of \(f \text \) In other words, a function is odd if performing a reflection about the \(y\)-axis and \(x\)-axis (doesn’t matter which is performed first) does not change the graph of the function.

## What is an odd function example?

What is an Odd Function? –

- Assume f to be a real-valued function of a variable that is real. The function f is odd when the equation is valid for all the values of x in a way that x and – x is present in the domain of the function f,
- -f(x) = f(-x)
- Or equivalently,
- f(x) + f(-x) = 0
- For example, f(x) = x 3 is an odd function, because for all value of x, -f(x) = f(-x).

### What’s the difference between odd and even?

What are even and odd numbers? – Even numbers are divisible by 2 without remainders. They end in 0, 2, 4, 6, or 8. Odd numbers are not evenly divisible by 2 and end in 1, 3, 5, 7, or 9. You can tell whether a number is odd or even regardless of how many digits it has by looking at the final digit.

## Is a linear function even or odd?

This linear function is symmetric about the origin and is an odd function : f ( x ) = f ( − x ). As shown earlier in the concept, this quadratic function is symmetric about the -axis and is an even function: f ( x ) = f ( − x ).

## What types of functions are even or odd?

How are you supposed to tell even and odd functions apart? An even function is one whose graph exhibits about the y -axis; an odd function is one whose graph exhibits symmetry about the origin. Which is a fancy way of saying that, if you split the graphs down the middle at the y -axis, an even function’s halves will mirror each other exactly, while an odd function’s halves with be upside-down of each other.

In order to “determine algebraically” whether a function is even, odd, or neither, you take the function and plug − x in for x, simplify, and compare the results with what you’d started with. If you end up with the exact same function that you started with (that is, if f (− x ) = f ( x ), so all of the signs are the same), then the function is even; if you end up with the exact opposite of what you started with (that is, if f (− x ) = − f ( x ), so all of the signs are switched), then the function is odd.

If the result is neither exactly the same nor exactly opposite (that is, if the result has neither all the same terms nor all the same terms but with opposite signs), then the function is neither even nor odd. Most functions, in fact, will be neither even nor odd.

#### Which of the 12 functions are even?

Even Functions: The squaring function and the absolute value function. Odd Functions: The identity function, the cubing function, the reciprocal function, the sine function. Neither: The square root function, the exponential function and the log function.

## Which function is an odd function?

A function is odd if, for each x in the domain of f, f(−x)=−f(x). Odd functions have 180° rotational symmetry about the origin.

## What is an example of a function that isn’t even or odd?

When we think “even and odd,” usually even and odd numbers are what come to mind. But what are even and odd functions? In today’s video, we will define even and odd functions and discuss how to identify them. Let’s begin by talking about even functions. Notice that the shape of this familiar parabola is visibly symmetric. The left and right sides of the plane are identical, just flipped. We can also show that this function is even algebraically, by evaluating at \(-x\), So our original function is \(f(x)=x^ \),

And we said that, if \(f(-x)\) is the same as \(f(x)\), then the function is even. So let’s evaluate at \(-x\), So, wherever there’s an \(x\), we’re gonna plug in \(-x\), So we have: \(f(-x)=(-x)^ \) Which, when you square a negative, it turns positive, so this is equal to: \(f(-x)=(-x)^ =x^ \) So in this case, \(f(-x)=f(x)\),

And because of our definition of even, this function, \(f(x)=x^ \), is even. Notice that if we add a constant to this function, it won’t affect the shape of the function, just raise or lower it on the plane. For example, this is the graph of \(f(x)=x^ +1\), Now let’s talk about what odd functions are like. Consider another function \(f(x)\), which we will once again evaluate at \(-x\), But this time, instead of looking for the same \(f(x)\) we started with, we want to see if \(f(-x)\) changes the sign of all terms in the function.

In other words, if \(f(-x)=-f(x)\), then the function is odd, Graphically, an odd function will appear the same when we rotate it by 180°, like flipping a page upside down, and it must pass through the origin. A few examples of odd functions are: \(f(x)=x\), \(f(x)=x^ \), and \(f(x)=sin(x)\) Let’s take a look at what’s going on here algebraically, using \(f(x)=x^ \) as an example.

So we’re gonna have our original function: \(f(x)=x^ \), And just like before, we’re gonna evaluate it at \(-x\), So we want \(f(-x)\), So anywhere we see an \(x\), we’re gonna plug in \(-x\), \(f(-x)=(-x)^ \) So \((-x)^ \), which is \(-x\cdot-x\cdot-x\), which means, since there are three negatives, our final answer’s gonna be \(-x^ \),

- F(-x)=-x^ \) So if you notice, \(f(x)\) is the opposite of \(f(-x)\),
- Each term, which in this case we only have one, is changed from positive to negative.
- So that means that this function is odd.
- Notice that if we were to add a constant to this function, it would no longer be odd.
- Remember that for odd functions, every term must change signs when evaluating at \(-x\),

That constant term would have no way to change sign, and we would see on the graph that the function would no longer pass through the origin. So we have now talked about definitions of both even and odd functions but before we go further, it’s important to clarify that some functions may be neither even nor odd! For example, take a look at the function \(f(x)=(x+1)^ \), From the graph we can see that this function doesn’t pass through the origin, so it can’t be odd. And it isn’t symmetric about the \(y\) -axis, so it isn’t even either. But we can also determine this algebraically. So our function is \(f(x)=(x+1)^ \), And remember, to determine if it’s even or odd, we want to evaluate it at \(-x\),

- So we have: \(f(-x)=(-x+1)^ \) Which we can write as: \(f(-x)=(-x+1)^ =(-x+1)(-x+1)\) ‘cause that’s what the squared means.
- And then from here we can FOIL.
- So: \(-x\cdot-x=x^ \) \(1\cdot-x=-x\) \(-x\cdot1=-x\) And, \(1\cdot1=1\) So we have: \(f(-x)=x^ -2x+1\) But remember, to determine even or odd, we have to compare this to our original function.

So let’s expand that. So for this, we have: \(f(x)=(x+1)(x+1)\) Which is: \(x⋅x=x^ \) \(1\cdot x=x\text x\cdot1=x,\text 2x\) \(1\cdot1=1\) \(f(x)=(x+1)(x+1)=x^ +2x+1\) Now, when we compare these two functions, we see that only one of the three terms ended up changing signs, so \(f(x)\) is not odd.

And since one term did change sign, \(f(x)\neq f(-x)\), so the function is not even either. Now that we’ve laid a groundwork for understanding even and odd functions, let’s talk about why we call them even and odd in the first place. Remember that even functions are the same when we evaluate them at \(+x\) and at \(-x\),

As we saw earlier, \(f(x)=x^ \) satisfies this property because anytime we square something, a positive value is returned, and therefore the sign of that term doesn’t change even if we plug a negative value in. The same is true when something is raised to the fourth power, or the sixth, and so on.

- Notice that constants do not change sign when we evaluate at \(-x\) either.
- That’s why we saw that the function \(f(x)=x^ +1\) was still even.
- As you can see, an even function will have even exponents,
- It may be unsurprising now that odd functions likewise will have odd exponents ! Remember that in order for a function to be odd, all terms must change sign when we evaluate at \(-x\),

Clearly, any term with \(x\) to the first power will change sign when we plug in a negative value of \(x\), In the same way, \(x\) to the third power, the fifth power, and so on will all change sign when we plug in a negative value for \(x\), As we mentioned earlier, when a term has an even power of \(x\), it will not change sign. Neither. This function is not symmetric about the \(y\) -axis, so it is not even. And even though it passes through the origin, it is not odd either because it would not appear the same if we were to rotate the image 180°. Let’s try another one. Is the function \(f(x)=\frac x^ -2x\) even, odd, or neither? Let’s look at each term.

- First, \(\frac x^ \) has an odd power of \(x\), meaning that the sign will change when evaluated at \(-x\),
- Similarly, the second term, \(-2x\), has an odd power of \(x\) and will also change sign.
- That means this function is odd! Let’s finish with a more conceptual question.
- We know that some functions may be neither even nor odd, but is it possible for a function to be both even and odd? Surprisingly, the answer is yes, but only for one function.

Can you think of what function that is? Remember that for even functions, \(f(-x)=f(x)\), and for odd functions, \(f(-x)=-f(x)\), The only way both of these can be satisfied is when \(f(x)=0\), \(f(-x)=f(x)\) \(and\) \(f(-x)=-f(x)\) As a quick recap, we can identify even and odd functions in the following ways: Graphically, even functions are symmetric about the \(y\) -axis.

- And they don’t have to pass through the origin.
- Though, odd functions must pass through the origin, and they will appear the same when viewed from a 180° rotation.
- Algebraically, even functions are the same when we evaluate at \(+x\) and at \(-x\),
- Odd functions will change signs across all terms when evaluated at \(-x\),

As a shortcut, if a function contains only even exponents of \(x\) (and may or may not have constants) then it is even. If a function has no constants and only odd exponents of \(x\), then it is odd. Now that we’ve covered everything and run through some examples, you should be pretty comfortable with identifying even and odd functions.

## Is 0 neither odd nor even?

Zero is an even number. In other words, its parity—the quality of aninteger being even or odd—is even. The simplest way to prove that zero iseven is to check that it fits the definition of ‘even’: it is an integermultiple of 2, specifically 0 × 2.

## Is 2.5 odd or even?

Standard Form –

- The standard form to represent the even number and odd numbers are as follows:
- Even Number = 2n
- Odd Number = 2n+1
- Where “n” can be any integer.

Zero is an even number because it obeys all the properties of the even number such as divisibility rule, and number line rule. The number 2.5 is neither even nor odd because it does not obey the property of even and odd numbers. If ” n” is an integer, the even numbers are represented in the form “2n”.

## How do you identify odd numbers?

Odd Numbers (Definition, Chart, Properties & Solved Examples) Odd numbers are the numbers that cannot be divided by 2 evenly. It cannot be divided into two separate integers evenly. If we divide an odd number by 2, then it will leave a remainder. The examples of odd numbers are 1, 3, 5, 7, etc.

## Is 67 odd or even?

67 (sixty-seven) is the natural number following 66 and preceding 68. It is an odd number.

#### What makes a function graph even?

A function is said to be an even function if its graph is symmetric with respect to the y-axis. For example, the function f graphed below is an even function. Verify this for yourself by dragging the point on the x-axis from right to left.

#### What makes a polynomial function even?

A polynomial is even if each term is an even function. A polynomial is odd if each term is an odd function. A polynomial is neither even nor odd if it is made up of both even and odd functions.

### What makes a rational function even?

Options: If the signs all stay the same or all change, f(-x) = f(x), then you have even or y-axis symmetry. If either the numerator or the denominator changes signs completely, f(-x)= -f(x) then you have odd, or origin symmetry.

#### What makes a trig function even?

Key Concepts –

- The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
- The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
- The six trigonometric functions can be found from a point on the unit circle. See Example,
- Trigonometric functions can also be found from an angle. See Example,
- Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See Example,
- A function is said to be even if \(f(−x)=f(x)\) and odd if \(f(−x)=−f(x)\).
- Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
- Even and odd properties can be used to evaluate trigonometric functions. See Example,
- The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
- Identities can be used to evaluate trigonometric functions. See Example and Example,
- Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See Example,
- The trigonometric functions repeat at regular intervals.
- The period \(P\) of a repeating function f f is the smallest interval such that \(f(x+P)=f(x)\) for any value of \(x\).
- The values of trigonometric functions of special angles can be found by mathematical analysis.
- To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See Example,