Problem 72 - Calculus for Biology and Medicine

Question

(a) Show that \(f(x)=\sqrt{x}, x \geq 0\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\)

Step-by-Step Solution

Answer

The function \( f(x) = \sqrt{x} \) is one-to-one, and its inverse is \( f^{-1}(x) = x^2 \) for \( x \geq 0 \). Graphically, \( f^{-1}(x) \) is the reflection of \( f(x) \) across the line \( y=x \).

Step 1: Showing that the function is one-to-one

To prove that the function \( f(x) = \sqrt{x} \) is one-to-one, we must show that if \( f(a) = f(b) \), then \( a = b \). Assume \( \sqrt{a} = \sqrt{b} \). Squaring both sides gives \( a = b \). Since this is true for any non-negative \( a \) and \( b \), the function \( f(x) = \sqrt{x} \) is one-to-one.

Step 2: Finding the inverse function

To find the inverse of \( f(x) = \sqrt{x} \), start by setting \( y = f(x) = \sqrt{x} \), and solve for \( x \) in terms of \( y \). This gives \( x = y^2 \), which means the inverse function is \( f^{-1}(x) = x^2 \). Since \( f(x) \) is defined for \( x \geq 0 \), the range of \( f(x) \) becomes the domain of \( f^{-1}(x) \). Therefore, the domain of \( f^{-1}(x) = x^2 \) is \( y \geq 0 \).

Step 3: Graphing the functions and the line y=x

Graph the function \( f(x) = \sqrt{x} \), which is a curve starting from the origin (0,0) and increasing slowly as \( x \) increases. Then, graph the inverse function \( f^{-1}(x) = x^2 \), which is a parabola opening upwards from the origin. Lastly, plot the line \( y = x \), which is a diagonal line through the origin with a 45-degree angle (slope of 1). When you reflect \( f(x) = \sqrt{x} \) across \( y = x \), you get the inverse \( f^{-1}(x) = x^2 \), illustrating the reflection property of inverse functions.

Key Concepts

Understanding One-to-One Functions

For a function to be considered one-to-one, each value in the domain is mapped to a unique value in the range. This means that no two different inputs (let's call them \(a\) and \(b\)) can produce the same output. In mathematical terms, a function \(f\) is one-to-one if and only if \(f(a) = f(b)\) implies \(a = b\). In our exercise, we have \(f(x) = \sqrt{x}\). To show this function is one-to-one:

• Assume \(f(a) = f(b)\).
• Then, \(\sqrt{a} = \sqrt{b}\).
• Square both sides, giving us \(a = b\).

This confirms that \(f(x) = \sqrt{x}\) is indeed one-to-one because no two different \(x\) values will produce the same result. This property is crucial when finding inverse functions.

Exploring Domain and Range

The domain of a function is all the possible input values (\(x\) values), while the range is all the possible output values (\(f(x)\) values). For our function \(f(x) = \sqrt{x}\):

• The domain is \(x \geq 0\), because square roots are defined only for non-negative numbers.
• The range is also \(f(x) \geq 0\), because square roots yield non-negative results.

Now, let's consider the inverse function \(f^{-1}(x) = x^2\):

• Here, the domain is derived from the range of the original function, so \(y \geq 0\).
• Meanwhile, the range of \(f^{-1}(x)\) contains all non-negative numbers, \(x \geq 0\).

Understanding domain and range helps ensure that our mappings between function and inverse remain accurate.

Graphing Functions and Their Inverses

Graphing functions helps visualize how they behave and relate to their inverses. For \(f(x) = \sqrt{x}\):

• This graph starts at the origin \((0,0)\), and moves slowly upwards as \(x\) increases.
• It's a gentle curve increasing from left to right.

Now, graph \(f^{-1}(x) = x^2\):

• This is a parabola that opens upwards from the origin.

To see the relationship between a function and its inverse, include the line \(y=x\):

• This line bisects the graph at a 45-degree angle.
• Reflecting \(f(x) = \sqrt{x}\) across this line will perfectly overlay \(f^{-1}(x) = x^2\).

Drawing these graphs gives you a clearer picture of how functions and their inverses are symmetrical around the line \(y=x\). This visualization emphasizes the reflective property of inverse functions in a graphical sense.