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For each equation below, determine if the function is Odd, Even, or Neither.
If this is a quiz or exam question, all steps and logic must be shown for full credit.
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Ask by Thornton Hammond. in the United States

Jan 23,2025

Question

Uploads: You must upload your work in the last question on the assessment. Your score is not valid unless
all work is uploaded on that question and can be viewed and read by your instructor.
Work: Unless a problem says otherwise, you may only use graphing calculators (ie Desmos) to check your
answers. All algebraic work and steps must be shown and be consistent with the appropriate readings.
Resources: These are the ONLY authorized resources you may use on this assessment:
For each equation below, determine if the function is Odd, Even, or Neither.
If this is a quiz or exam question, all steps and logic must be shown for full credit.
Select an answer
Select an answer
Select an answer

Ask by Thornton Hammond. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To determine whether the functions $$ f(x) $$, $$ g(x) $$, and $$ h(x) $$ are odd, even, or neither, we will use the definitions of even and odd functions:

- A function $$ f(x) $$ is **even** if $$ f(-x) = f(x) $$ for all $$ x $$ in the domain of $$ f $$.
- A function $$ f(x) $$ is **odd** if $$ f(-x) = -f(x) $$ for all $$ x $$ in the domain of $$ f $$.

Let's analyze each function step by step.

### 1. Function $$ f(x) = x^2 + 4 $$

**Step 1:** Calculate $$ f(-x) $$:
$$
f(-x) = (-x)^2 + 4 = x^2 + 4
$$

**Step 2:** Compare $$ f(-x) $$ with $$ f(x) $$:
$$
f(-x) = x^2 + 4 = f(x)
$$

Since $$ f(-x) = f(x) $$, the function $$ f(x) $$ is **even**.

### 2. Function $$ g(x) = (x - 4)^2 $$

**Step 1:** Calculate $$ g(-x) $$:
$$
g(-x) = (-x - 4)^2 = (-(x + 4))^2 = (x + 4)^2
$$

**Step 2:** Expand $$ g(-x) $$:
$$
g(-x) = x^2 + 8x + 16
$$

**Step 3:** Compare $$ g(-x) $$ with $$ g(x) $$:
$$
g(x) = (x - 4)^2 = x^2 - 8x + 16
$$

Since $$ g(-x) \neq g(x) $$ and $$ g(-x) \neq -g(x) $$, the function $$ g(x) $$ is **neither** even nor odd.

### 3. Function $$ h(x) = \frac{4}{x} $$

**Step 1:** Calculate $$ h(-x) $$:
$$
h(-x) = \frac{4}{-x} = -\frac{4}{x}
$$

**Step 2:** Compare $$ h(-x) $$ with $$ h(x) $$:
$$
h(-x) = -h(x)
$$

Since $$ h(-x) = -h(x) $$, the function $$ h(x) $$ is **odd**.

### Summary of Results:
- $$ f(x) = x^2 + 4 $$ is **even**.
- $$ g(x) = (x - 4)^2 $$ is **neither**.
- $$ h(x) = \frac{4}{x} $$ is **odd**.

These conclusions can be summarized as follows:
- $$ f(x) $$: Even
- $$ g(x) $$: Neither
- $$ h(x) $$: Odd
- $$ f(x) = x^2 + 4 $$ is **even**. - $$ g(x) = (x - 4)^2 $$ is **neither**. - $$ h(x) = \frac{4}{x} $$ is **odd**.

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