Question

precalculus
name:
even and odd functions
determine whether ( f ) is even, odd or neither even nor odd.

1. ( f(x)=5 x^{3}+2 x )
2. ( f(x)=|x|-3 )
3. ( f(x)=3 x^{4}+2 x^{2}-5 )
4. ( f(x)=7 x^{5}-4 x^{3} )

Explanation:

1. For \( f(x)=5x^{3}+2x \)

Step1: Find \( f(-x) \)

Substitute \( x \) with \( -x \) in \( f(x) \).
\( f(-x)=5(-x)^{3}+2(-x)=-5x^{3}-2x=-(5x^{3} + 2x)\)

Step2: Compare with \( f(x) \)

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

2. For \( f(x)=\vert x\vert - 3 \)

Step1: Find \( f(-x) \)

Substitute \( x \) with \( -x \) in \( f(x) \).
\( f(-x)=\vert -x\vert - 3=\vert x\vert - 3\)

Step2: Compare with \( f(x) \)

Since \( f(-x)=f(x) \) (because \( f(x)=\vert x\vert - 3 \)), the function is even.

3. For \( f(x)=3x^{4}+2x^{2}-5 \)

Step1: Find \( f(-x) \)

Substitute \( x \) with \( -x \) in \( f(x) \).
\( f(-x)=3(-x)^{4}+2(-x)^{2}-5 = 3x^{4}+2x^{2}-5\)

Step2: Compare with \( f(x) \)

Since \( f(-x)=f(x) \) (because \( f(x)=3x^{4}+2x^{2}-5 \)), the function is even.

4. For \( f(x)=7x^{5}-4x^{3} \)

Step1: Find \( f(-x) \)

Substitute \( x \) with \( -x \) in \( f(x) \).
\( f(-x)=7(-x)^{5}-4(-x)^{3}=-7x^{5}+4x^{3}=-(7x^{5}-4x^{3})\)

Step2: Compare with \( f(x) \)

Since \( f(-x)=-f(x) \) (because \( f(x)=7x^{5}-4x^{3} \)), the function is odd.

Answer:

1. Odd function
2. Even function
3. Even function
4. Odd function