Question

1. (8 points) four functions are given below. either the function is defined explicitly, or the entire graph of the function is shown.

for each, decide whether it is an even function, an odd function, or neither. circle the appropriate response below each function.

graph 1
odd even neither

graph 2
odd even neither

h(x) = -x^4 - 3x^2
odd even neither

f(x) = -2x^7 - x^3
odd even neither

Explanation:

Step1: Analyze the first graph

The graph is not symmetric about the y-axis or origin.

$$\text{Neither}$$

Step2: Analyze the second graph

The graph is symmetric with respect to the y-axis.

$$\text{Even}$$

Step3: Evaluate the function H(x)

Substitute \(-x\) into \(H(x) = -x^4 - 3x^2\).

$$H(-x) = -(-x)^4 - 3(-x)^2 = -x^4 - 3x^2 = H(x)$$

Step4: Evaluate the function f(x)

Substitute \(-x\) into \(f(x) = -2x^7 - x^3\).

$$f(-x) = -2(-x)^7 - (-x)^3 = 2x^7 + x^3 = -f(x)$$

Answer:

1. Graph 1 (top-left): Neither
2. Graph 2 (top-right): Even
3. \(H(x) = -x^4 - 3x^2\): Even
4. \(f(x) = -2x^7 - x^3\): Odd