Solved on Feb 27, 2024

Identify the odd functions from the given expressions: $f(x)=-2 x^{2}+5$ , $f(x)=5 x$ , $f(x)=-5 x^{5}-2 x^{3}+4 x$ , $f(x)=2 x^{3}+3 x^{2}+4 x+3$ , $f(x)=-3 x^{3}+6 x$ .

STEP 1

Assumptions
1. A function $f(x)$ is considered an odd function if it satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
2. We are given multiple functions to test for oddness.
3. We will apply the property to each function to determine if it is odd.

STEP 2

Let's start by testing the first function $f(x)=-2x^{2}+5$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -2(-x)^{2}+5$

STEP 3

Simplify the expression by squaring $-x$ .
$f(-x) = -2x^{2}+5$

STEP 4

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-2x^{2}+5) = 2x^{2}-5$

STEP 5

Since $f(-x) \neq -f(x)$ , the function $f(x)=-2x^{2}+5$ is not an odd function.

STEP 6

Next, let's test the second function $f(x)=5x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = 5(-x)$

STEP 7

Simplify the expression.
$f(-x) = -5x$

STEP 8

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(5x) = -5x$

STEP 9

Since $f(-x) = -f(x)$ , the function $f(x)=5x$ is an odd function.

STEP 10

Now, let's test the third function $f(x)=-5x^{5}-2x^{3}+4x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -5(-x)^{5}-2(-x)^{3}+4(-x)$

STEP 11

Simplify the expression by applying the odd powers.
$f(-x) = 5x^{5}+2x^{3}-4x$

STEP 12

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-5x^{5}-2x^{3}+4x) = 5x^{5}+2x^{3}-4x$

STEP 13

Since $f(-x) = -f(x)$ , the function $f(x)=-5x^{5}-2x^{3}+4x$ is an odd function.

STEP 14

Next, let's test the fourth function $f(x)=2x^{3}+3x^{2}+4x+3$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = 2(-x)^{3}+3(-x)^{2}+4(-x)+3$

STEP 15

Simplify the expression by applying the powers.
$f(-x) = -2x^{3}+3x^{2}-4x+3$

STEP 16

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(2x^{3}+3x^{2}+4x+3) = -2x^{3}-3x^{2}-4x-3$

STEP 17

Since $f(-x) \neq -f(x)$ , the function $f(x)=2x^{3}+3x^{2}+4x+3$ is not an odd function.

STEP 18

Finally, let's test the fifth function $f(x)=-3x^{3}+6x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -3(-x)^{3}+6(-x)$

STEP 19

Simplify the expression by applying the odd powers.
$f(-x) = 3x^{3}-6x$

STEP 20

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-3x^{3}+6x) = 3x^{3}-6x$

STEP 21

Since $f(-x) = -f(x)$ , the function $f(x)=-3x^{3}+6x$ is an odd function.
The odd functions from the given list are:
$f(x)=5x$
$f(x)=-5x^{5}-2x^{3}+4x$
$f(x)=-3x^{3}+6x$

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Identify the odd functions from the given expressions: f(x)=−2x2+5f(x)=-2 x^{2}+5f(x)=−2x2+5, f(x)=5xf(x)=5 xf(x)=5x, f(x)=−5x5−2x3+4xf(x)=-5 x^{5}-2 x^{3}+4 xf(x)=−5x5−2x3+4x, f(x)=2x3+3x2+4x+3f(x)=2 x^{3}+3 x^{2}+4 x+3f(x)=2x3+3x2+4x+3, f(x)=−3x3+6xf(x)=-3 x^{3}+6 xf(x)=−3x3+6x.

STEP 1

STEP 2

Let's start by testing the first function f(x)=−2x2+5f(x)=-2x^{2}+5f(x)=−2x2+5.Apply the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).f(−x)=−2(−x)2+5f(-x) = -2(-x)^{2}+5f(−x)=−2(−x)2+5

STEP 3

Simplify the expression by squaring −x-x−x.f(−x)=−2x2+5f(-x) = -2x^{2}+5f(−x)=−2x2+5

STEP 4

Now, compare f(−x)f(-x)f(−x) with −f(x)-f(x)−f(x).−f(x)=−(−2x2+5)=2x2−5-f(x) = -(-2x^{2}+5) = 2x^{2}-5−f(x)=−(−2x2+5)=2x2−5

STEP 5

Since f(−x)≠−f(x)f(-x) \neq -f(x)f(−x)=−f(x), the function f(x)=−2x2+5f(x)=-2x^{2}+5f(x)=−2x2+5 is not an odd function.

STEP 6

Next, let's test the second function f(x)=5xf(x)=5xf(x)=5x.Apply the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).f(−x)=5(−x)f(-x) = 5(-x)f(−x)=5(−x)

STEP 7

Simplify the expression.f(−x)=−5xf(-x) = -5xf(−x)=−5x

STEP 8

Now, compare f(−x)f(-x)f(−x) with −f(x)-f(x)−f(x).−f(x)=−(5x)=−5x-f(x) = -(5x) = -5x−f(x)=−(5x)=−5x

STEP 9

Since f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), the function f(x)=5xf(x)=5xf(x)=5x is an odd function.

STEP 10

Now, let's test the third function f(x)=−5x5−2x3+4xf(x)=-5x^{5}-2x^{3}+4xf(x)=−5x5−2x3+4x.Apply the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).f(−x)=−5(−x)5−2(−x)3+4(−x)f(-x) = -5(-x)^{5}-2(-x)^{3}+4(-x)f(−x)=−5(−x)5−2(−x)3+4(−x)

STEP 11

Simplify the expression by applying the odd powers.f(−x)=5x5+2x3−4xf(-x) = 5x^{5}+2x^{3}-4xf(−x)=5x5+2x3−4x

STEP 12

Now, compare f(−x)f(-x)f(−x) with −f(x)-f(x)−f(x).−f(x)=−(−5x5−2x3+4x)=5x5+2x3−4x-f(x) = -(-5x^{5}-2x^{3}+4x) = 5x^{5}+2x^{3}-4x−f(x)=−(−5x5−2x3+4x)=5x5+2x3−4x

STEP 13

Since f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), the function f(x)=−5x5−2x3+4xf(x)=-5x^{5}-2x^{3}+4xf(x)=−5x5−2x3+4x is an odd function.

STEP 14

Next, let's test the fourth function f(x)=2x3+3x2+4x+3f(x)=2x^{3}+3x^{2}+4x+3f(x)=2x3+3x2+4x+3.Apply the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).f(−x)=2(−x)3+3(−x)2+4(−x)+3f(-x) = 2(-x)^{3}+3(-x)^{2}+4(-x)+3f(−x)=2(−x)3+3(−x)2+4(−x)+3

STEP 15

Simplify the expression by applying the powers.f(−x)=−2x3+3x2−4x+3f(-x) = -2x^{3}+3x^{2}-4x+3f(−x)=−2x3+3x2−4x+3

STEP 16

Now, compare f(−x)f(-x)f(−x) with −f(x)-f(x)−f(x).−f(x)=−(2x3+3x2+4x+3)=−2x3−3x2−4x−3-f(x) = -(2x^{3}+3x^{2}+4x+3) = -2x^{3}-3x^{2}-4x-3−f(x)=−(2x3+3x2+4x+3)=−2x3−3x2−4x−3

STEP 17

Since f(−x)≠−f(x)f(-x) \neq -f(x)f(−x)=−f(x), the function f(x)=2x3+3x2+4x+3f(x)=2x^{3}+3x^{2}+4x+3f(x)=2x3+3x2+4x+3 is not an odd function.

STEP 18

Finally, let's test the fifth function f(x)=−3x3+6xf(x)=-3x^{3}+6xf(x)=−3x3+6x.Apply the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).f(−x)=−3(−x)3+6(−x)f(-x) = -3(-x)^{3}+6(-x)f(−x)=−3(−x)3+6(−x)

STEP 19

Simplify the expression by applying the odd powers.f(−x)=3x3−6xf(-x) = 3x^{3}-6xf(−x)=3x3−6x

STEP 20

Now, compare f(−x)f(-x)f(−x) with −f(x)-f(x)−f(x).−f(x)=−(−3x3+6x)=3x3−6x-f(x) = -(-3x^{3}+6x) = 3x^{3}-6x−f(x)=−(−3x3+6x)=3x3−6x

STEP 21

Since f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), the function f(x)=−3x3+6xf(x)=-3x^{3}+6xf(x)=−3x3+6x is an odd function.The odd functions from the given list are:f(x)=5xf(x)=5xf(x)=5xf(x)=−5x5−2x3+4xf(x)=-5x^{5}-2x^{3}+4xf(x)=−5x5−2x3+4xf(x)=−3x3+6xf(x)=-3x^{3}+6xf(x)=−3x3+6x

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Identify the odd functions from the given expressions: $f(x)=-2 x^{2}+5$ , $f(x)=5 x$ , $f(x)=-5 x^{5}-2 x^{3}+4 x$ , $f(x)=2 x^{3}+3 x^{2}+4 x+3$ , $f(x)=-3 x^{3}+6 x$ .

Let's start by testing the first function $f(x)=-2x^{2}+5$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -2(-x)^{2}+5$

Simplify the expression by squaring $-x$ .
$f(-x) = -2x^{2}+5$

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-2x^{2}+5) = 2x^{2}-5$

Since $f(-x) \neq -f(x)$ , the function $f(x)=-2x^{2}+5$ is not an odd function.

Next, let's test the second function $f(x)=5x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = 5(-x)$

Simplify the expression.
$f(-x) = -5x$

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(5x) = -5x$

Since $f(-x) = -f(x)$ , the function $f(x)=5x$ is an odd function.

Now, let's test the third function $f(x)=-5x^{5}-2x^{3}+4x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -5(-x)^{5}-2(-x)^{3}+4(-x)$

Simplify the expression by applying the odd powers.
$f(-x) = 5x^{5}+2x^{3}-4x$

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-5x^{5}-2x^{3}+4x) = 5x^{5}+2x^{3}-4x$

Since $f(-x) = -f(x)$ , the function $f(x)=-5x^{5}-2x^{3}+4x$ is an odd function.

Next, let's test the fourth function $f(x)=2x^{3}+3x^{2}+4x+3$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = 2(-x)^{3}+3(-x)^{2}+4(-x)+3$

Simplify the expression by applying the powers.
$f(-x) = -2x^{3}+3x^{2}-4x+3$

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(2x^{3}+3x^{2}+4x+3) = -2x^{3}-3x^{2}-4x-3$

Since $f(-x) \neq -f(x)$ , the function $f(x)=2x^{3}+3x^{2}+4x+3$ is not an odd function.

Finally, let's test the fifth function $f(x)=-3x^{3}+6x$ .
Apply the property $f(-x) = -f(x)$ .
$f(-x) = -3(-x)^{3}+6(-x)$

Simplify the expression by applying the odd powers.
$f(-x) = 3x^{3}-6x$

Now, compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(-3x^{3}+6x) = 3x^{3}-6x$

Since $f(-x) = -f(x)$ , the function $f(x)=-3x^{3}+6x$ is an odd function.
The odd functions from the given list are:
$f(x)=5x$
$f(x)=-5x^{5}-2x^{3}+4x$
$f(x)=-3x^{3}+6x$