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Chapter 5: Problem 52

Graph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of \(f\) and of \(f^{-1}\). $$f(x)=3-x^{2}, x \geq 0$$

Short Answer

Expert verified
Domain of \(f\): \(x \geq 0\); Range of \(f\): \(0 \leq f(x) \leq 3\). Domain of \(f^{-1}\): \(0 \leq x \leq 3\); Range of \(f^{-1}\): \(x \geq 0\).

Step by step solution

01

Understand the Function

The function given is \(f(x) = 3 - x^2\) with the domain \(x \geq 0\). This means the function is defined for all non-negative values of \(x\).
02

Graph the Function

Using a graphing calculator, graph the function \(f(x) = 3 - x^2\) only for \(x \geq 0\). This would give you a downward-opening parabola starting from \(x=0\).
03

Find the Inverse of the Function

To find the inverse function, solve the equation \(y = 3 - x^2\) for \(x\). Swap \(x\) and \(y\) and solve for \(y\). This gives \(x = 3 - y^2\), thus \(y = \sqrt{3 - x}\). Therefore, \(f^{-1}(x) = \sqrt{3 - x}\).
04

Graph the Inverse Function

Graph the inverse function \(f^{-1}(x) = \sqrt{3 - x}\) using the graphing calculator. This will be a half-parabola opening to the right, starting from \(x=0\).
05

Identify the Domains and Ranges

For the function \(f(x) = 3 - x^2\), since \(x \geq 0\), the domain is restricted to \(x \geq 0\) and the range is \(0 \leq f(x) \leq 3\). For the inverse function \(f^{-1}(x) = \sqrt{3 - x}\), the domain is \(0 \leq x \leq 3\) and the range is \(x \geq 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Domain and Range
To get a handle on any function, it's essential to understand its domain and range.
The domain of a function is all the possible input values (x-values) that will produce a valid output. The range is all the possible output values (y-values).
For the function given, \(f(x) = 3 - x^2\), with the constraint \(x geq 0\), the domain is \(x geq 0\). This tells us we're only considering non-negative x-values.
Because \(f(x) = 3 - x^2\) is a downward-opening parabola, the highest point is when \(x=0\) (which gives \(f(0)=3\)), so the range of \(f(x)\) is \(0 leq f(x) leq 3\).
Graphing Functions
Graphing functions helps visualize the behavior of functions and their inverses. In our case, we need to graph \(f(x) = 3 - x^2\) and its inverse.
When graphing \(f(x)\), remember it's a downward parabola starting from \(x=0\). Using a graphing calculator simplifies this process.
For the inverse function, solve \((y = 3 - x^2)\) for x, swapping x and y. This gives \(x = 3 - y^2\), so \(y = qrt{3 - x}\).
The inverse function \(f^{-1}(x) = qrt{3 - x}\) will be a right-opening half-parabola. With inverse graphing tools, you can compare both the original and inverse functions visually.
Parabolas
A parabola is a symmetrical, curved shape that looks like an arch.
The standard form for a parabola's equation is \(y = ax^2 + bx + c\). In our function, \(f(x) = 3 - x^2\) is a downward parabola because of the negative sign in front of \(x^2\).
Key characteristics include the vertex (the highest or lowest point on the parabola) and the direction it opens (upward or downward).
For our function \(f(x)\), the vertex is (0, 3), and it opens downward.
For the inverse function \(f^{-1}(x) = qrt{3 - x}\), it forms a right-opening half-parabola starting from x=0.
Understanding these properties is crucial for graphing and interpreting both the function and its inverse accurately.

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