Question

Determine whether the function is one-to-one. $$ f(x)=-x^{4}+16 $$

Short answer

The function \(f(x) = -x^4 + 16\) is not one-to-one, as its first derivative \(f'(x) = -4x^3\) reveals that the function is increasing for \(x < 0\) and decreasing for \(x > 0\), meaning it doesn't have a continuous increasing or decreasing behavior throughout its domain.

Step-by-step solution

Find the first derivative

We will first find the first derivative of the function \(f(x) = -x^4 + 16\), which gives us information about the function's increasing and decreasing behavior. Differentiate \(f(x)\) with respect to \(x\): \[f'(x) = \frac{d}{dx}(-x^4 + 16)\] Using the power rule, we have: \[f'(x) = -4x^3\]

Analyze the first derivative

Analyze the first derivative, \(f'(x) = -4x^3\), to determine where the function is increasing or decreasing. Since the function is a cubic function, as \(x\) becomes more negative, \(f'(x)\) becomes more positive, and as \(x\) becomes more positive, \(f'(x)\) becomes more negative. This means that the function \(f(x) = -x^4 + 16\) is increasing for \(x < 0\) and decreasing for \(x > 0\).

Conclusion

We can conclude that the function \(f(x) = -x^4 + 16\) is increasing for \(x < 0\), and then decreasing for \(x > 0\). Since the function doesn't have a continuous increasing or decreasing behavior throughout its domain, this function is not one-to-one.

Key concepts

First Derivative Test

Understanding the first derivative test is essential when analyzing the behavior of functions. It involves taking the derivative of a function and using it to determine where the function is increasing or decreasing. By setting the first derivative equal to zero, you can find critical points, which are potential maxima, minima, or points of inflection.

In our exercise, to apply this test to the function f(x) = -x^4 + 16, we first computed its derivative. Critical points occur when the derivative is zero or undefined. However, for our cubic function, there are no points where f'(x) is zero, meaning there are no local maxima or minima. Instead, analyzing the sign of the first derivative, which is -4x^3, we note that it changes sign as x crosses from negative to positive, indicating a change in the function's behavior from increasing to decreasing.

Increasing and Decreasing Functions

A function is deemed increasing if, as x increases, f(x) also increases. Conversely, it is decreasing if, as x increases, f(x) decreases. To determine this, one must look at the sign of the first derivative. If f'(x) is positive over an interval, the function is increasing there; if it's negative, the function is decreasing.

In the given function f(x) = -x^4 + 16, by observing the first derivative, we concluded that the function is increasing on the interval where x < 0. This is because f'(x) is positive in that interval. Likewise, for interval where x > 0, the first derivative is negative, indicating that the function is decreasing. The transition from increasing to decreasing reinforces that the function cannot be one-to-one, since a one-to-one function would not have such a reversal in direction.

Power Rule Differentiation

The power rule is a quick method to differentiate functions of the form f(x) = x^n, where n is any real number. According to this rule, the derivative of such a function is f'(x) = nx^(n-1). It simplifies the differentiation process significantly.

When we applied the power rule to differentiate our function, f(x) = -x^4 + 16, we multiplied the exponent 4 by the coefficient of the x term, which was -1, and then subtracted one from the exponent. Consequently, the derivative was f'(x) = -4x^3. The power rule is especially handy for polynomial functions, as it allows for quick and systematic calculation of derivatives, which are critical when assessing the function's increasing or decreasing intervals as well as for the first derivative test.