Question3. $f(x)=x^{4}-4 x^{2}$

Studdy Solution

STEP 1

1. We are given the function $f(x) = x^4 - 4x^2$ .
2. The task is to analyze or manipulate the function as needed.

STEP 2

1. Identify the type of function and its characteristics.
2. Simplify the function if possible.
3. Determine the roots of the function.
4. Analyze the behavior of the function.

STEP 3

Identify the type of function:
The function $f(x) = x^4 - 4x^2$ is a polynomial function of degree 4. It is a quartic function, which is symmetric about the y-axis because it only contains even powers of $x$ .

STEP 4

Simplify the function:
Factor the expression:
$f(x) = x^4 - 4x^2$
Factor out the common term $x^2$ :
$f(x) = x^2(x^2 - 4)$
Recognize that $x^2 - 4$ is a difference of squares:
$f(x) = x^2(x - 2)(x + 2)$

STEP 5

Determine the roots of the function:
Set the factored form equal to zero:
$x^2(x - 2)(x + 2) = 0$
Solve for $x$ :
1. $x^2 = 0$ gives $x = 0$ .
2. $x - 2 = 0$ gives $x = 2$ .
3. $x + 2 = 0$ gives $x = -2$ .
The roots of the function are $x = 0, 2, -2$ .

STEP 6

Analyze the behavior of the function:
1. The function is symmetric about the y-axis.
2. The roots indicate that the function crosses the x-axis at $x = 0, 2, -2$ .
3. As $x \to \infty$ or $x \to -\infty$ , $f(x) \to \infty$ because the leading term $x^4$ dominates.
The function $f(x) = x^4 - 4x^2$ is fully analyzed with roots at $x = 0, 2, -2$ and symmetry about the y-axis.

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