Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=\sqrt{x-1}\)

Step-by-Step Solution

Answer

The graph of \(f(x) = \sqrt{x - 1}\) is a half-parabola shape positioned to the right of the y-axis, starting from the point (1, 0). The domain of the function is \([1, \infty)\).

Step 1: Define the Domain

Understand the domain of the function. Since inside the square root, there is x - 1, the function is defined for x such that \(x - 1 \geq 0\). So, \(x \geq 1\). The domain is \([1, \infty)\).

Step 2: Identify Key Points

Identify the key points of the function. By plugging in some x-values from the domain into the function we get corresponding y-values. Let's plug in 1, 2, and 4. If x = 1, \(f(1) = \sqrt{1-1} = 0\), if x = 2, \(f(2) = \sqrt{2-1} = 1\), if x = 4, \(f(4) = \sqrt{4-1} = \sqrt{3}\). This gives us points (1, 0), (2, 1) and (4, \(\sqrt{3}\)).

Step 3: Plot Points and Draw the Graph

Start by plotting the points (1, 0), (2, 1), and (4, \(\sqrt{3}\)). The graph of \(f(x) = \sqrt{x - 1}\) will be a half-parabola shape positioned to the right of the y-axis, and the curve will pass through the points plotted. Remember, the curve will not go to the left of the point (1,0) since our domain starts at x = 1.

Key Concepts

Domain of Functions

When analyzing a function, the domain is a big deal because it tells us all the input values (x-values) that we can use without breaking any mathematical rules. For square root functions like \(f(x) = \sqrt{x - 1}\), it's crucial to ensure that whatever is inside the square root symbol is zero or more because square roots of negative numbers aren't real, at least in basic math.
In this function, we have \(x - 1\) inside our square root. To find the domain, we solve the inequality \(x - 1 \geq 0\). By adding 1 to both sides, we get \(x \geq 1\), meaning the domain is all numbers from 1 up. We write this as \([1, \infty)\).

• Start by setting the inside of the square root \(\geq 0\).
• Solve for \(x\).
• Write down all the legal x-values in interval notation.

This gives us a clear picture of where our function lives on the x-axis.

Key Points Identification

After finding the domain, identifying key points is our next step. These are special spots on the graph that help us see its shape and direction.
For \(f(x) = \sqrt{x - 1}\), we pick values of \(x\) from the domain \([1, \infty)\) and find out what each corresponds to in \(y\) when plugged into the function. Key points make plotting the graph manageable.

• Choose easy x-values like 1, 2, and 4.
• Calculate corresponding y-values. For example, \(f(1) = \sqrt{1-1} = 0\).
• These calculations give us points: (1, 0), (2, 1), and (4, \(\sqrt{3}\)).

These points are critical because they act like signposts on our graph, showing us the path the curve will follow.

Square Root Functions

Square root functions bring a unique touch to graphing and are known for their distinct shape. The classic half-parabola look of a square root function is due to its mathematical nature.
The basic form is \(\sqrt{x}\), which mostly sticks to the positive side of the x-axis, since negative inputs aren't allowed. With our specific function \(f(x) = \sqrt{x - 1}\), it's simply shifted one unit to the right because of the \(-1\) before the square root.
These functions naturally start at the point where the inside of the square root is zero, emphasizing the importance of domain restrictions. As \(x\) increases, \(y\) values grow positively but more slowly over time, creating that lovely curve.

• Recognize the starting point of the graph due to the domain restrictions.
• Understand that the curve grows but flattens out as it moves along the x-axis.
• This rightward half-parabola gives square root functions its iconic shape.

Visualizing this graph helps cement our understanding of square root functions and lets us accurately sketch its unique, smooth path.