How to find if a trigonometric function is even, odd or neither and its period, witho

Sep 2014
82
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UK
I am asked to find whether the functions are odd, even, neither and their period, if they have one, of the following functions:

f(x) = sin(x) + cos(x) and f(x) = sin(x)cos(x).

I know that even if f(x) = f(-x) and odd if f(-x) = -f(x) but all those fuzzy functions come up like for f(x) = sin(x) + cos(x)
f(-x) = sin(-x) + cos(-x), how am I suppose to know if f(-x) is the same as f(x) without putting in random values of x?

Help please, thanks.
 
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greg1313

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Is $\sin(x)$ even? Odd? Neither? Is $\cos(x)$ even? Odd? Neither?
 
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skeeter

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look at the graphs of $y=\sin{x}$, $y=\cos{x}$, and $y=\tan{x}$ ... you should know that the graphs of even & odd functions display a specific kind of symmetry.

once you've determined whether the three basic trig functions are even, odd, or neither, then have a look at a couple of simpler examples ...

$x^2 \cdot x^3$ is an even function times an odd function.

$x^2+x^3$ is an even function + an odd function.

what do you think about the resulting function formed by their sum & their product as far as being even, odd, or neither?


can you extend this idea to $\sin{x}+\cos{x}$ and $\sin{x} \cdot \cos{x}$ ?
 
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Sep 2014
82
1
UK
I done the following:
\(\displaystyle x^2 + x^2\) is even + even and it equals even.
\(\displaystyle x^2 + x^3\) is even + odd and it equals even.
\(\displaystyle x^3 + x^3\) is odd + odd and it equals even.

\(\displaystyle x^2 * x^2\) is Even x Even and it is equal to Neither since some x give an even number and others give an odd number.
\(\displaystyle x^2 * x^3\) is Even x Odd and it is equal to Neither again.
\(\displaystyle x^3 * x^3\) is Odd x Odd and it is equal to Neither again.

y = sinx + cosx is Odd + Even so it should be Even but its neither..?

y= sinxcosx is Odd x Even so it should be neither but its odd..?

I don't get it.. I have done some numbers for each function, e.g. an Odd + Even I would do x =2,3,4,5 and get an even number all the time. So I don't get how its neither.

Edit:

I done it algebraically and I do now see how you get neither for even + odd and odd for odd x even. But shouldn't it give me same answers if I plugged in numbers..? A bit confused why im getting different algebraic results to when I plug numbers in.
 
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skeeter

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I done the following:
\(\displaystyle x^2 + x^2\) is even + even and it equals even.
\(\displaystyle x^2 + x^3\) is even + odd and it equals even. $\color{red}{no}$
\(\displaystyle x^3 + x^3\) is odd + odd and it equals even.$\color{red}{no}$

\(\displaystyle x^2 * x^2\) is Even x Even and it is equal to Neither since some x give an even number and others give an odd number. $\color{red}{no}$
\(\displaystyle x^2 * x^3\) is Even x Odd and it is equal to Neither again. $\color{red}{no}$
\(\displaystyle x^3 * x^3\) is Odd x Odd and it is equal to Neither again.$\color{red}{no}$
I don't think you really understand the concept ... odd and even functions are not the same as odd and even numbers.
 
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greg1313

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Given $f(x)=\sin(x)+\cos(x)$ and $g(x)=\sin(x)\cos(x)$ and knowing whether $\sin(x)$ and $\cos(x)$ are even or odd, all you have to do is plug in $-x$ and see how the target function behaves.

Is $f(-x)=-f(x)$? Is $f(-x)=f(x)$? Do the same for $g(x)$.
 
Sep 2014
82
1
UK
Given $f(x)=\sin(x)+\cos(x)$ and $g(x)=\sin(x)\cos(x)$ and knowing whether $\sin(x)$ and $\cos(x)$ are even or odd, all you have to do is plug in $-x$ and see how the target function behaves.

Is $f(-x)=-f(x)$? Is $f(-x)=f(x)$? Do the same for $g(x)$.
I know what you're saying and I have done that.
f(x) = sin x + cos x
f(-x) = sin(-x) + cos(-x)
-f(x) = -(sin(x) + cos(x)) = -sin(x) - cos(x)

f(x) = sin(x)cos(x)
f(-x) = sin(-x)cos(-x)
-f(x) = -sin(x)cos(x)

Since f(x) = sin(x)cos(x) is odd, it would mean that f(-x) = -f(x) so sin(-x)cos(-x) = -sin(x)cos(x).

But how can I tell sin(-x)cos(-x) = -sin(x)cos(x) without knowing what the graphs of sin(x)cos(x), -sin(x)cos(x) and sin(-x)cos(-x) look like?

Anyway, I do understand the method of determining whether combined trig functions are even, odd or neither by looking at what even + even, even + odd and so on gives me and determining through that, I just don't understand how you'd do it otherwise.
 
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skipjack

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You made a sign error somewhere.

I'll assume throughout that x is real and that g(x) and h(x) are defined for all real x.

Consider s(x) ≡ g(x) + h(x) and p(x) ≡ g(x)h(x).

If g(x) and h(x) are both even, i.e., g(-x) ≡ g(x) and h(-x) ≡ h(x),
s(-x) ≡ g(-x) + h(-x) ≡ g(x) + h(x) ≡ s(x) and p(x) ≡ g(-x)h(-x) ≡ g(x)h(x) ≡ p(x), so s(x) and p(x) are both even.

If g(x) and h(x) are both odd, i.e., g(-x) ≡ -g(x) and h(-x) ≡ -h(x),
s(-x) ≡ g(-x) + h(-x) ≡ -g(x) - h(x) ≡ -s(x) and p(-x) ≡ g(-x)h(-x) ≡ (-g(x))(-h(x)) ≡ p(x), so s(x) is odd and p(x) is even.

If g(x) is even and h(x) is odd, i.e., g(-x) ≡ g(x) and h(-x) ≡ -h(x),
p(-x) ≡ g(-x)h(-x) ≡ g(x)(-h(x)) ≡ -p(x), so p(x) is odd.

Now try the questions below about g(x) and h(x).

(a) If g(x) is both an even function and an odd function, what function is it?

(b) Is h(x) + h(-x) an even function?

(c) Can h(x) - h(-x) be an even function?

(d) Is g(x²) an even function?

(e) If g(x³) is an odd function, is g(x) an odd function?

(f) If g(x) is an even function and h(x) is an odd function, is g(h(x)) an even function?

(g) If g(x) is an even function and h(x) is an odd function, is h(g(x)) an even function?

(h) Given that (x, y) is any point on the ellipse with equation x² + 4y² = 1, is y an even function of x?
 
Sep 2014
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(a) If g(x) is both an even function and an odd function, what function is it?

(b) Is h(x) + h(-x) an even function?

(c) Can h(x) - h(-x) be an even function?

(d) Is g(x²) an even function?

(e) If g(x³) is an odd function, is g(x) an odd function?

(f) If g(x) is an even function and h(x) is an odd function, is g(h(x)) an even function?

(g) If g(x) is an even function and h(x) is an odd function, is h(g(x)) an even function?

(h) Given that (x, y) is any point on the ellipse with equation x² + 4y² = 1, is y an even function of x?
(a) Self inverse?

I don't know how to do literally like the rest of them. For (b), I don't know if h(x) is even or odd, neither do I know if h(-x) is even or odd, so how am I suppose to answer this question?

I do get how to manipulate the functions kind of... I found that:

Even+Even = Even since \(\displaystyle x^2 + x^2 = 2x^2\)
Even + Odd = Neither since \(\displaystyle x^2 + x^3 \)
Odd + Odd = Even since \(\displaystyle x^3 + x^3 = 2x^3\)

Same for multiplication:

Even x Even = Even \(\displaystyle x^2 * x^2 = x^4\)
Even x Odd = Odd \(\displaystyle x^2 * x^3 = x^5\)
Odd x Odd = Even \(\displaystyle x^3 * x^3 = x^6\)

This allowed me to answer the two questions with the trigonometric functions but I don't get the questions that you just asked me.
 
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skeeter

Math Team
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(a) Self inverse?

I don't know how to do literally like the rest of them. For (b), I don't know if h(x) is even or odd, neither do I know if h(-x) is even or odd, so how am I suppose to answer this question?

I do get how to manipulate the functions kind of... I found that:

Even+Even = Even since \(\displaystyle x^2 + x^2 = 2x^2\)
Even + Odd = Neither since \(\displaystyle x^2 + x^3 \)
Odd + Odd = Even since \(\displaystyle x^3 + x^3 = 2x^3\) $\color{red}{no}$

Same for multiplication:

Even x Even = Even \(\displaystyle x^2 * x^2 = x^4\)
Even x Odd = Odd \(\displaystyle x^2 * x^3 = x^5\)
Odd x Odd = Even \(\displaystyle x^3 * x^3 = x^6\)

This allowed me to answer the two questions with the trigonometric functions but I don't get the questions that Ive just asked me.
$f(x)=2x^3$ is odd, because $f(-x)=2(-x)^3 = -2x^3 = -f(x)$
 
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