Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(x)=x^{2}+x-6, x \geq 2\)

Step-by-Step Solution

Answer

The function \( f(x) = x^2 + x - 6 \), \( x \geq 2 \), is strictly increasing and thus has an inverse.

Step 1: Understand the Function's Domain

The function is defined as \( f(x) = x^2 + x - 6 \) for \( x \geq 2 \). This means we only consider the behavior of the function for values starting from 2 and going upwards.

Step 2: Find the Derivative of the Function

To determine if the function is strictly monotonic, we start by finding its derivative. The derivative of \( f(x) = x^2 + x - 6 \) is given by applying the power rule to each term: \( f'(x) = 2x + 1 \).

Step 3: Analyze the Sign of the Derivative

Since \( f'(x) = 2x + 1 \), we can analyze its sign over the domain \( x \geq 2 \). Substitute \( x = 2 \) into the derivative to confirm: \( f'(2) = 2(2) + 1 = 5 \), which is positive. Because \( 2x + 1 \) is strictly increasing, \( f'(x) > 0 \) for all \( x \geq 2 \).

Step 4: Conclude the Strict Monotonicity and Existence of Inverse

Since \( f'(x) > 0 \) for all \( x \geq 2 \), the function \( f(x) = x^2 + x - 6 \) is strictly increasing. A strictly increasing function over its domain implies it is one-to-one, and therefore, \( f \) has an inverse.

Key Concepts

Monotonic Functions

A monotonic function is one that consistently moves in one direction as its input increases, meaning it never changes direction between increasing and decreasing. To put it simply, the function either goes up entirely or goes down entirely.
Monotonic functions are crucial in determining the existence of inverse functions. Why? An inverse function exists when each output is uniquely connected to one input, and this happens naturally when a function is monotonic and doesn't double back on itself.

• **Increasing Functions** keep getting larger as you move along their graph.
• **Decreasing Functions** get smaller.

The original function example from the exercise, with its domain starting at 2 for the function\( f(x)=x^{2}+x-6 \), is strictly monotonic because it consistently rises with every increase in \( x \). This ensures it doesn't loop back on itself, thus making the function invertible.

Derivative

The derivative is like a magic tool in mathematics. It helps you see how a function changes or slopes at any given point.
For the function \( f(x)=x^{2}+x-6 \), the derivative \( f'(x)=2x+1 \) tells us the rate of change of the function's output with respect to its input.
Derivatives are especially useful when determining whether a function is increasing or decreasing:

• If the derivative, \( f'(x) \), is positive for all \( x \) in your domain, then the function is increasing.
• If \( f'(x) \) is negative, the function is decreasing.

In our exercise, checking the derivative gives \( f'(x) = 2x + 1 \). For \( x \geq 2 \), \( f'(x) \) is always positive, meaning the function rises constantly in this range. This helps us conclude the function is strictly monotonic.

Strictly Increasing Functions

A function is strictly increasing if, as one variable grows, so does the output, consistently without plateauing or decreasing at any point in its domain.
The function \( f(x)=x^{2}+x-6 \) within the constraint \( x \geq 2 \) is a perfect example. It never backtracks.
When function shows such steady growth, each input corresponds to one unique output. This "one-to-one" characteristic is essential for defining an inverse function.
The rigorous test shows that since \( f'(x) = 2x + 1 \) is positive over the entire domain \( x \geq 2 \), the function is strictly increasing. There's no turning back or leveling out, straightforwardly meeting the conditions for an inverse.