Problem 50 - Thomas Calculus

Question

Determine conditions on the constants \(a, b, c,\) and \(d\) so that the rational function $$ f(x)=\frac{a x+b}{c x+d} $$ has an inverse.

Step-by-Step Solution

Answer

The conditions are \( c \neq 0 \) and \( ad-bc \neq 0 \).

Step 1: Understand the conditions for a function to have an inverse

A function has an inverse if it is one-to-one, meaning that different inputs map to different outputs. For a rational function, this often implies that the function must be strictly monotonous, or the denominator is not zero for any real number.

Step 2: Determine when the rational function is defined for all x

The rational function \( f(x) = \frac{a x + b}{c x + d} \) is undefined when the denominator is zero. Thus, the function is defined for all \( x \) except when \( cx + d = 0 \). Therefore, a restriction is needed so that \( c eq 0 \).

Step 3: Check for one-to-one property using derivative

To check if the function is one-to-one, compute its derivative and ensure it never changes sign. The derivative \( f'(x) \) is given by the quotient rule: \[ f'(x) = \frac{ad-bc}{(cx+d)^2}. \] For \( f(x) \) to be one-to-one, \( ad-bc eq 0 \). This means the slope does not change sign, indicating a consistently increasing or decreasing function.

Step 4: Conclusion on conditions

From the calculations, the conditions for \( f(x) \) to have an inverse are: \( c eq 0 \) to ensure the function is defined, and \( ad-bc eq 0 \) to ensure the function is one-to-one.

Key Concepts

Inverse Function

An inverse function is essentially a function that "reverses" the effect of the original function. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying \( f \) followed by \( f^{-1} \) (or vice versa) will bring you back to your starting point.

• If \( f(x) = y \), then \( f^{-1}(y) = x \).
• This reversal property is only possible if each output matches uniquely to one input.

If a function can establish a clear path to reverse, it can be called invertible or having an inverse function. For a rational function, establishing this requires checking specific conditions, such as the absence of zero in the denominator across its range and examining its behavior for uniqueness, which brings us to the next point.

One-to-One Function

A core requirement for a function to have an inverse is that it needs to be one-to-one. This essentially means each element of the domain maps to a unique element in the range. Here's how you can think about it:

• No two different inputs should produce the same output.
• Visually, this means any horizontal line crosses the graph at most once.

For the given rational function \( f(x) = \frac{a x + b}{c x + d} \), ensuring the function is one-to-one involves a couple of requirements:- Check the derivative's sign.- The slope, calculated through the derivative \( \frac{ad-bc}{(cx+d)^2} \), should never change (no zero-crossings), meaning \( ad-bc eq 0 \). Notably, confirming a one-to-one relationship here is crucial for defining an inverse.

Derivative

The derivative provides the mathematical tool to examine how a function behaves. It tells us about the slope of a function, but here it serves another purpose: determining monotonicity, which helps establish the one-to-one nature.For our rational function, the derivative is derived using the quotient rule:\[f'(x) = \frac{ad-bc}{(cx+d)^2}\]

• The numerator \( ad-bc \) plays a crucial role. If it equals zero, it means the slope could change, violating one-to-one property.
• If \( ad-bc eq 0 \), the slope format ensures that the function is always increasing or decreasing uniformly.

Thus, the derivative helps check essential conditions that impact both the one-to-oneness and the definition of an inverse.

Monotonicity

Monotonicity is about how a function consistently moves in one direction—either always increasing or always decreasing. A monotonic function has no turning points or flattening peaks where it could change direction. This kind of behavior verifies that the function is one-to-one.For the function \( f(x) = \frac{a x + b}{c x + d} \):

• We determine monotonicity by looking at the sign of the derivative.
• The derivative \( \frac{ad-bc}{(cx+d)^2} \) must be either entirely positive or negative across its domain.

Therefore, ensuring \( ad-bc eq 0 \) confirms the function does not reverse its course, maintaining a consistent direction vital for the invertibility potential. Thus, knowing a function is monotonic solidifies it as a candidate for having an inverse, provided other conditions are also met.