Question

The function \(f\) is one-to-one. (a) Find its inverse function \(f^{-1}\) and check your answer. (b) Find the domain and the range of \(f\) and \(f^{-1}\). (c) Graph \(f, f^{-1},\) and \(y=x\) on the same coordinate axes. $$ f(x)=x^{2}+9, \quad x \geq 0 $$

Short answer

solve for y, replace x Check domain Graph points

Step-by-step solution

Rewrite the function for finding the inverse

Given the function is one-to-one: Monotonocity is evident from the graph below $$\begin{aligned} f(x)=x 8 x 9,quad y cleq0 \but => $$

Isolate and Solve for x

Replace f(x) with y: $$y = x^2 + 9$$ Replace y and x to solve for x $$x =y^2 + 9$$ \reflects the inverse function equation

Define the Domain of f(x)

Considering the function representation: $$x 9$$ \begin eq $[ 7, x \rightarrow \frac {\textrm domain f range left {y \rightarrow x} sqrt{y+ 9} >\frac x }$$ Read the domain easier to identify

Define the Range of f(x)

Since x⩾0 , this domain is downward$$ bounded the intervals of which Created the domain ()

Find Domain and Range of f^{-1}(x)

Reflect the function Identify the domain and range observse equals to$$ inverse f^{-1}(x): x=f(x)=sqrt x=9$$Monotocity

Graph f(x), f^{-1}(x) and the line y=x simultaneously

Create points and plots xy-axis Representation on points and coordinates

Key concepts

one-to-one function

A one-to-one function, also known as an injective function, means that every output value is matched by one unique input value. In other words, no two different inputs will produce the same output. This is crucial because only one-to-one functions have inverses. For the given function, \(f(x) = x^2 + 9\), it's defined for \(x \geq 0\). This ensures the function is one-to-one since for every non-negative \(x\), the value of \(x^2 + 9\) is unique.

Here are some key points to identify and confirm if a function is one-to-one:

• Check if each value in the range corresponds to exactly one value in the domain.
• If the function passes the Horizontal Line Test, it's a one-to-one function. This means that any horizontal line passing through the graph will intersect it at most once.
• For our function, because it's increasing for \(x \geq 0\), it passes the Horizontal Line Test, confirming it's one-to-one.

domain and range

The domain of a function consists of all possible input values (\(x\)), while the range consists of all possible output values (\(y\)). For our function \(f(x) = x^2 + 9\), the domain and range are:

• Domain of \(f(x)\): Given \(f(x)\) is defined for \(x \geq 0\), the domain of \(f(x)\) is \([0, \, \infty)\).
• Range of \(f(x)\): For every non-negative \(x\), \(x^2\) ranges from \(0\) to infinity. So, \(f(x) = x^2 + 9\) ranges from \(9\) to infinity. Therefore, the range is \([9, \, \infty)\).

To find the domain and range of the inverse function \(f^{-1}(x)\):

• Since \(f(x)\) is one-to-one, it has an inverse. The inverse can be found by swapping \(x\) and \(y\) and solving for \(y\). For our function, \(y = x^2 + 9\), swapping gives \(x = y^2 + 9\), which simplifies to \(y = \sqrt{x-9}\).
• Domain of \(f^{-1}(x)\): As \(x\) must be greater than or equal to \(9\) (since \(x \geq 0\) initially and \(x - 9 \geq 0\)), the domain of the inverse function is \([9, \, \infty)\).
• Range of \(f^{-1}(x)\): For \(\sqrt{x-9}\), since \(x-9 \geq 0\), \(y\) ranges from \(0\) to infinity, making the range \([0, \, \infty)\).

graphing functions

When graphing functions, it's essential to clearly understand what the graph represents and how the inverse function relates to the original function. Here’s how you can approach graphing \(f(x)\), \(f^{-1}(x)\), and the line \(y=x\):

• Graph of \(f(x)\): Plot the function \(f(x) = x^2 + 9\) for \(x \geq 0\). This will be a parabola shifted upward by 9 units.
• Graph of \(f^{-1}(x)\): Plot the inverse function \(f^{-1}(x) = \sqrt{x-9}\). This graph starts at \(x = 9\) and rises linearly.
• Line \(y=x\): This line will help illustrate the reflection property of the inverse function. The graph of \(f^{-1}(x)\) is a reflection of \(f(x)\) across the line \(y=x\).

For a clear graph:

• Use a consistent scale for both axes.
• Label important points, such as the intercepts and points where the function graph intersects the line \(y=x\).
• Ensure your graph reflects the domains and ranges accurately.
• Double-check the plot points to confirm they follow the function equations correctly.

Through graphing, you'll visually see how each function behaves and how they relate to each other.