1. Classify each function as odd, even, or neither.
a. $f (x) = 3 x^{4} + 3$
b. $f (x) = x^{3} - 4 x$
c. $f (x) = \frac{1}{x^{2} + 1}$
d. $f (x) = x^{2} + x - 3$

Answer

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Answer

1d. f(-x)=(-x)^{2}+(-x)-3 = x^{2}-x-3 \neq f(x) \neq -f(-x), \Rightarrow \text{neither}

Steps

Step 1 ：1a. f(-x)=3(-x)^{4}+3 = 3x^{4}+3, f(x)=f(-x) \Rightarrow \text{even}

Step 2 ：1b. f(-x)=(-x)^{3}-4(-x) = -x^{3}+4x, f(x)=-f(-x) \Rightarrow \text{odd}

Step 3 ：1c. f(-x)=\frac{1}{(-x)^{2}+1} = \frac{1}{x^{2}+1}, f(x)=f(-x) \Rightarrow \text{even}

Step 4 ：1d. f(-x)=(-x)^{2}+(-x)-3 = x^{2}-x-3 \neq f(x) \neq -f(-x), \Rightarrow \text{neither}