Question

Determine where the graph of \(f\) is below the graph of g by solving the inequality \(f(x) \leq g(x) .\) Graph \(f\) and g together. \(f(x)=x^{4}-1\) \(g(x)=x-1\)

Short answer

The graph of \(f(x) \) is below \( g(x) \) between \ x \ in \ [0, 1] \.

Step-by-step solution

- Set up the inequality

We need to determine where the function \(f(x)\) is below or equal to the function \(g(x)\). Set up the inequality: \[x^4 - 1 \leq x - 1.\]

- Simplify the inequality

Rearrange the inequality to find the points of intersection by moving all terms to one side: \[x^4 - x \leq 0.\]

- Factor the inequality

Factor the left side to find the critical points: \[x^4 - x = x(x^3 - 1) = x(x-1)(x^2 + x + 1).\]

- Identify the critical points

Set each factor equal to zero to find the critical points: \[x = 0, x-1 = 0 \Rightarrow x = 1\] The quadratic term \(x^2 + x + 1\) does not have real roots.

- Test intervals around critical points

Test intervals around the critical points \(x = 0\) and \(x = 1\) to determine where the inequality holds: \[-\text{infinity to 0, 0 to 1, and 1 to \infty}.\]

- Evaluate the polynomial signs

For each interval, determine if \(x(x - 1)(x^2 + x + 1) \leq 0.\) The inequality holds between \([0,1]\)

- Graph the functions

Graph both \(f(x) = x^4 - 1\) and \(g(x) = x - 1\) on the same coordinate axis to visually confirm where \(f(x)\) is below \(g(x)\).

Key concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to nonnegative integer exponents and their coefficients. They're fundamental in algebra.For the exercise above, we deal with two functions:

• Function 1: f(x) = x^4 - 1, a polynomial of degree 4.
• Function 2: g(x) = x - 1, a polynomial of degree 1 (a linear function).

Polynomial functions can be simple like g(x), but they can also be complicated like f(x). Understanding them requires learning about their degrees, leading coefficients, and general shape. The degree of a polynomial provides insights into its behavior at large values and the number of roots it can have. For example, f(x) with degree 4 means, at most, it has 4 roots and behaves similarly to x^4 for large values. To solve inequalities involving polynomials, we often need to simplify and factor them.

Factoring

Factoring is a method used to break down polynomials into simpler pieces or 'factors' that can be multiplied together to give the original polynomial.

In the given exercise, we factored the expression from the inequality:

• Original inequality after rearrangement: x^4 - x ≤ 0
• Factored form: x(x - 1)(x^2 + x + 1) ≤ 0

Factoring helps to reveal the roots or critical points of the polynomial. These points are where the polynomial intersects the x-axis. For example, setting each factor equal to zero, we find critical points: x = 0 and x = 1. The third factor, x^2 + x + 1, does not yield real roots but affects the polynomial's sign in different intervals. Factoring is essential in analyzing and solving polynomial inequalities.

Testing Intervals

After factoring, we need to determine where the polynomial satisfies the inequality. This involves testing intervals around the critical points.

In our example, we have identified critical points at 0 and 1. We then choose intervals based on these points:

• Interval 1: (-∞, 0)
• Interval 2: (0, 1)
• Interval 3: (1, ∞)

We pick test points (like -1, 0.5, and 2) within each interval to substitute back into the factored inequality. If the polynomial result satisfies the inequality (being less than or equal to zero), then the interval is part of the solution. Here, testing reveals that only the interval [0, 1] works. This method ensures we correctly capture where the function meets the inequality conditions.

Graphical Analysis

Graphing functions provides a visual understanding of where one function is below or above another. It makes polynomial inequalities clearer.

For this exercise, we graph both functions: f(x) = x^4 - 1 and g(x) = x - 1.
When plotted on the same coordinate system, we can see where the graphs intersect and where one is above or below the other.Graphical analysis confirms our solution derived from algebraic methods. It visually verifies that f(x) stays below g(x) between [0, 1].

Key points to note when graphing:

• Identify intersections by solving the equality f(x) = g(x).
• Determine the shape and curvature from the polynomial's degree and leading coefficient.
• Label critical points and intervals where one function surpasses the other.

Using graphing technology, like graphing calculators or software, can be very helpful in this process. Analyzing the graphs strengthens the understanding and solves the inequality comprehensively.