Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{x-1}\) \(x=1\)

Short answer

The domain of the function \(f(x)=\sqrt{x-1}\) is \([1, +\infty)\), its range is \([0, +\infty)\), and the value of the function at \(x = 1\) is 0.

Step-by-step solution

Determine the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the square root function \(f(x)=\sqrt{x-1}\), the value inside the square root (called the radicand) must be non-negative (i.e., it must be either positive or zero). So, we set \(x-1 \geq 0\). Solving for \(x\), we get \(x \geq 1\). Therefore, the domain of the function is \([1, +\infty)\).

Determine the Range

The range of a function is the set of all possible output values (y-values). Because of the nature of the square root function, where the output of a square root is always non-negative (i.e., positive or zero), the range is \([0, +\infty)\).

Evaluate the Function at \(x = 1\)

To find the value of the function at \(x = 1\), substitute \(1\) into the function: \(f(1)=\sqrt{1-1} = \sqrt{0} = 0\).

Key concepts

Evaluating Functions

When you evaluate a function, you're basically finding out what its output is for a particular input. Imagine you have a machine that takes in numbers and spits out results according to a specific rule. That rule is your function. For instance, with the function given in our exercise, \(f(x)=\sqrt{x-1}\), when you want to evaluate this function at \(x=1\), you're looking for the output when the input is 1.

You simply replace the variable \(x\) with 1 in the equation and then calculate: \(f(1)=\sqrt{1-1} = \sqrt{0} = 0\). So when \(x=1\), the function's output, or its value, is 0. This process can be used for any value of \(x\) that is part of the domain, which leads us into our next important concept: domain of a function.

Square Root Functions

A square root function is a special kind of beast. It includes the square root symbol, which represents a mathematical operation that asks the question: What number, multiplied by itself, gives me what's under the root?

For the function \(f(x)=\sqrt{x-1}\), the expression under the root is \(x-1\). Now, since squaring a negative number gives you a positive one (think \((-2) \times (-2) = 4\)), the reverse doesn't work — you can't have the square root of a negative number if you're only dealing with real numbers. That's why the domain of square root functions is so important. And it's also why the output of a square root function is always zero or positive, which affects the range.

Function Definition

A function is like a specific instruction for turning inputs (usually \(x\) values) into outputs (usually \(y\) or \(f(x)\) values). But not all instructions are valid for every possible input. Think of it like this: some ingredients can't go into certain recipes because they would ruin the dish. Similarly, the domain tells us which inputs are 'safe' for the function's recipe and will produce an actual output.

In our exercise, \(f(x)\) is defined as \(\sqrt{x-1}\). The \(\sqrt{}\) symbol means it's a square root function, which imposes certain restrictions. These restrictions, as mentioned before, are part of why we need to know the domain and range of a function — to understand what inputs work and what outputs we can expect.

Inequalities in Algebra

Inequalities are all about relationships — they tell us how numbers compare to each other. They come in various forms, like '<', '>', '\(\leq\)', and '\(\geq\)'. When you're working with functions, especially square root functions, inequalities help you figure out the domain.

Back to our example, \(x-1 \geq 0\) is an inequality that tells us the domain of \(f(x)=\sqrt{x-1}\). In order to keep the number under the square root non-negative, \(x\) must be greater than or equal to 1, hence the \(\geq\) symbol. This detail is crucial because it guides you on which \(x\) values you can plug into your function without breaking the mathematical 'laws' of square roots.