Determine where the graph of f is on or below the graph of g by solving the inequality

f left parenthesis x right parenthesis less than or equals g left parenthesis x right parenthesis .f(x)≤g(x).

Graph f and g together.

​f(x)equals=x Superscript 4 Baseline minus 8x4−8

​g(x)equals=2 x squared2x2

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Part 1

The graph of f is on or below the graph of g over

enter your response here.

​(Type your answer in interval​ notation.)

We are tasked with solving the inequality \( f(x) \leq g(x) \), where the functions are defined as follows:

- \( f(x) = x^4 - 8 \)
- \( g(x) = 2x^2 \)

### Step 1: Set up the inequality
We want to solve:

\[
x^4 - 8 \leq 2x^2
\]

### Step 2: Rearrange the inequality
Subtract \( 2x^2 \) from both sides to set the inequality to 0:

\[
x^4 - 2x^2 - 8 \leq 0
\]

This is a quartic inequality that we will solve by factoring.

### Step 3: Let \( y = x^2 \)
This substitution transforms the quartic expression into a quadratic form:

\[
y^2 - 2y - 8 \leq 0
\]

### Step 4: Solve the quadratic inequality
First, solve the related equation:

\[
y^2 - 2y - 8 = 0
\]

Use the quadratic formula to find the solutions:

\[
y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}
\]
\[
y = \frac{2 \pm \sqrt{4 + 32}}{2}
\]
\[
y = \frac{2 \pm \sqrt{36}}{2}
\]
\[
y = \frac{2 \pm 6}{2}
\]
\[
y = 4 \quad \text{or} \quad y = -2
\]

Since \( y = x^2 \geq 0 \), we discard \( y = -2 \). Thus, \( y = 4 \) is the valid solution.

### Step 5: Factor the quadratic
Now that we know the roots, we can factor the quadratic:

\[
y^2 - 2y - 8 = (y - 4)(y + 2)
\]

Thus, the inequality becomes:

\[
(y - 4)(y + 2) \leq 0
\]

### Step 6: Solve the inequality for \( y \)
To solve \( (y - 4)(y + 2) \leq 0 \), we analyze the sign changes of the expression by testing intervals based on the roots \( y = 4 \) and \( y = -2 \).

The critical points divide the \( y \)-axis into three intervals:
1. \( y < -2 \)
2. \( -2 \leq y \leq 4 \)
3. \( y > 4 \)

For each interval:
- When \( y < -2 \), the product \( (y - 4)(y + 2) > 0 \).
- When \( -2 \leq y \leq 4 \), the product \( (y - 4)(y + 2) \leq 0 \).
- When \( y > 4 \), the product \( (y - 4)(y + 2) > 0 \).

Thus, the solution to the inequality in terms of \( y \) is \( -2 \leq y \leq 4 \).

### Step 7: Translate back to \( x \)
Since \( y = x^2 \), we now solve:

\[
-2 \leq x^2 \leq 4
\]

Since \( x^2 \geq 0 \), the lower bound \( -2 \leq x^2 \) is always true. So, we only need to solve \( x^2 \leq 4 \), which gives:

\[
-2 \leq x \leq 2
\]

### Step 8: Conclusion
The graph of \( f(x) \) is on or below the graph of \( g(x) \) over the interval \( [-2, 2] \).

### Final Answer:
The graph of \( f \) is on or below the graph of \( g \) over the interval:

\[
\boxed{[-2, 2]}
\]

### Graphing
If you graph both functions, you will see that \( f(x) = x^4 - 8 \) and \( g(x) = 2x^2 \) intersect at \( x = -2 \) and \( x = 2 \), with \( f(x) \leq g(x) \) between these points.

---

Do you want more details on how the graphs of these functions behave, or do you have any other questions?

### Related Questions:
1. What is the difference between solving a quadratic and quartic inequality?
2. How do we factor higher-degree polynomials like quartics?
3. Why do we use interval testing to solve inequalities?
4. How can you determine the intersection points of two functions graphically?
5. What real-world situations can be modeled by inequalities like this?

### Tip:
Always check your intervals carefully when solving polynomial inequalities. Graphing the function can provide a visual check for accuracy.

Determine where the graph of f is on or below the graph of g by solving the inequality f(x) ≤ g(x). Graph f and g together. f(x) = x^4 - 8 and g(x) = 2x^2.

Explore how to solve the quartic inequality f(x) = x^4 - 8 ≤ g(x) = 2x^2 with step-by-step explanations. This problem involves factoring, applying the quadratic formula, and understanding polynomial inequalities. Ideal for students in Grades 10-12.

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