Problem 35 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=\sqrt{x-1} $$

Step-by-Step Solution

Answer

Domain: \([1, \infty)\); Range: \([0, \infty)\)

Step 1: Input the Function into the Graphing Calculator

Enter the function \(f(x) = \sqrt{x-1}\) into a graphing calculator or graphing tool. Ensure that the calculator is set to an appropriate window to view the curve effectively. Observe the shape of the graph, which is typically a horizontal parabola starting at the point where the function is defined.

Step 2: Determine the Domain of the Function

For the function \(f(x) = \sqrt{x-1}\), remember that the expression under the square root must be non-negative. Solve the inequality \(x - 1 \geq 0\) to find that \(x \geq 1\). Therefore, the domain of the function is all real numbers \(x\) such that \(x \geq 1\). In interval notation, this is \([1, \infty)\).

Step 3: Determine the Range of the Function

From the graph, observe the set of possible \(y\)-values the function can output. Since the square root function \(\sqrt{x-1}\) outputs values starting from 0 and increasing without bound, the range of \(f\) is \(y \geq 0\). In interval notation, the range is \([0, \infty)\).

Key Concepts

Graphing Calculator

To analyze mathematical functions, one of the most useful tools is the graphing calculator. It allows you to plot graphs of different functions to visually understand their behavior. Inputting the function into a graphing calculator provides a quick visual assessment of important qualities like the domain and range. When dealing with the function \( f(x) = \sqrt{x-1} \), it is critical to set up the graphing calculator correctly. Ensure the window or viewing area is adjusted properly to capture the essential parts of the graph.
This function begins at the point (1,0) and extends rightward, resembling a rising curve. The graphing calculator helps identify that the function doesn't exist for \( x < 1 \). This observation is crucial for discerning the domain and range, which are needed to fully understand the behavior of the function.

Square Root Function

The square root function, denoted as \( f(x) = \sqrt{x-1} \), is a type of radical function. It is defined only for values of \( x \) where the expression inside the square root is non-negative. For this function, \( x - 1 \geq 0 \) must hold true. Solving for \( x \), we find that \( x \geq 1 \). This means the function is only defined for \( x \) starting at 1 and increasing, which is vital in determining its domain.

• Domain: Values of \( x \) starting from 1 and going to infinity.
• The function is not defined for any number less than 1.

The shape of \( \sqrt{x-1} \) results in a graph that is limited to non-negative outputs. This arises because a square root cannot produce a negative result from a real number input when extending upwards from zero.

Interval Notation

Interval notation is a mathematical shorthand that describes subsets of real numbers. When analyzing functions like \( f(x) = \sqrt{x-1} \), interval notation efficiently communicates the domain and range.
For the domain, the condition \( x \geq 1 \) means the function is defined starting from 1 to positive infinity. In interval notation, this is expressed as \( [1, \infty) \). The bracket \([\) used with 1 indicates that 1 is included in the domain, and the parenthesis \()\) with infinity indicates that infinity is not a specific endpoint but a direction in the positive x-axis.

• Domain of \( f(x) = \sqrt{x-1} \) is \([1, \infty)\).
• Range of the same function is \([0, \infty)\).

Interval notation also clarifies the range of the function, which is the set of all possible output values. Since \( \sqrt{x-1} \) starts at 0 and increases, the outputs cover all non-negative numbers, represented in interval notation as \([0, \infty)\). This precise form of notation simplifies communication of these mathematical ideas.