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Chapter 5: Problem 8

\(f(x)=\sqrt{x-3}+3\)

Short Answer

Expert verified
The domain of the function is \([3, +\infty)\), the range of the function is \([3, +\infty)\), and a point of interest is (3,3).

Step by step solution

01

Determine and Understand the Domain

The domain of a function is the set of all possible input values (i.e., values of x) which will make the function work and will output real numbers. Here, as the function has a square root, it needs to be highlighted that inside the square root should be a non-negative value. Hence, \(x - 3\) must be greater than or equal to zero. Therefore, \(x\) should be greater than or equal to \(3\). After solving, the domain of the function is obtained, which is \([3, +\infty)\).
02

Determine and Understand the Range

The range of a function is the set of all possible output values (i.e., values of f(x)), which we can get after substituting all the possible values of \(x\) from its domain. Looking at the function, it should be noted that square root value cannot be negative and the smallest possible value it can give is zero, when \(x = 3\). Therefore, the range of this function will be \([3, +\infty)\) as well.
03

Identify Points of Interest

Points of interest for the function could be where the function crosses the x-axis (which doesn't happen here as the range starts from 3), or any other relevant point. Here, the point (3, 3) appears to be of particular interest because if you put \(x = 3\), you will get \(f(x) = 3\). This is the minimum value of the function and any other real values of \(x\) will generate values of \(f(x)\) greater than '3'.

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Most popular questions from this chapter

Let \(g\) be a reflection in the \(y\)-axis, followed by a translation 1 unit right of the graph of \(f(x)=2 \sqrt[3]{x-1}\).

\(h(x)=-\sqrt[4]{x}\)

\(f(x)=2 \sqrt{x}, g(x)=f(x+3)\)

\(|3 x+2|=5\)

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-x^3+3 $$
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