Problem 8 - Precalculus

Question

In Exercises \(1-12\), sketch the graph of the given function. State the domain of the function, identify any intercepts and test for symmetry. $$ f(x)=\sqrt{x-2} $$

Step-by-Step Solution

Answer

Domain: \([2, \infty)\); x-intercept at (2,0); No symmetry; Half-parabola starting at (2,0).

Step 1: Identify Domain

The function is defined for values of \(x\) where the expression inside the square root, \(x-2\), is non-negative. Set the inequality \(x-2 \geq 0\), which simplifies to \(x \geq 2\). Thus, the domain of \(f(x)\) is \([2, \infty)\).

Step 2: Calculate Intercepts

To find the x-intercept, set \(f(x) = 0\): \(\sqrt{x-2} = 0\). Solving gives \(x = 2\). Therefore, the x-intercept is at the point \((2, 0)\). There is no y-intercept since the domain starts at \(x = 2\).

Step 3: Test for Symmetry

Check if \(f(x) = f(-x)\) for even symmetry or \(f(-x) = -f(x)\) for odd symmetry. Substituting \(-x\) into the function gives \(f(-x) = \sqrt{-x - 2}\), which is undefined within the domain of \(f(x)\). Therefore, the function has no symmetry.

Step 4: Sketch the Graph

Start plotting the graph from the point \((2, 0)\) since this is the starting point of the domain. The graph increases slowly as \(x\) increases, represented by a half-parabola above the x-axis. The curve is only defined for \(x \geq 2\).

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values (here known as the values of \(x\)) for which the function is defined. To determine the domain of any square root function, you must ensure the expression inside the square root is non-negative.
This is because the square root of a negative number isn't defined in real numbers.
For the function \(f(x) = \sqrt{x-2}\), the term inside the square root is \(x-2\). Therefore, you solve the inequality \(x-2 \geq 0\). This simplifies to \(x \geq 2\).
This tells us that our domain is \([2, \infty)\), meaning \(x\) can be any number from 2 onwards.

This ensures the function is defined and will produce real, valid outputs.
Understanding the domain helps avoid errors in calculations because it clearly shows where the function is applicable.

X-Intercept

The x-intercept of a function is the point where the graph intersects the x-axis.
At this point, the value of \(y\) or \(f(x)\) is zero, and finding it involves solving for \(x\) when the function itself is equal to zero.
For \(f(x) = \sqrt{x-2}\), set \(f(x) = 0\):
\[ \sqrt{x-2} = 0 \]
Squaring both sides removes the square root:
\[ x - 2 = 0 \]
Add 2 to both sides to solve for \(x\):
\[ x = 2 \]
This means the x-intercept occurs at the point \((2, 0)\). There is no y-intercept here since as \(x\) approaches values less than 2, the function is not defined.
An x-intercept is crucial for understanding where the function touches or crosses the horizontal axis.

Symmetry

Symmetry in a function's graph involves the idea that one part of the graph is a mirror image of another.
It can either be even symmetry (mirror about the y-axis) or odd symmetry (rotational symmetry around the origin).
To test for symmetry in \(f(x) = \sqrt{x-2}\), we look at two conditions:

• For even symmetry: \(f(x) = f(-x)\)
• For odd symmetry: \(f(-x) = -f(x)\)

Substitute \(-x\) into the function:
\[ f(-x) = \sqrt{-x-2} \]
The expression \(\sqrt{-x-2}\) is undefined for any \(x\) where \(-x\) creates a negative under the square root.
This immediately tells us that the function has no symmetry because \(f(-x)\) doesn't satisfy conditions for calculated symmetry.
Understanding symmetry helps predict the function's behavior and check for errors in graphing.

Square Root Function

Square root functions are a type of radical function where the variable is under the square root.
They generally produce a curve that starts from a certain point and extends indefinitely.
For the function \(f(x) = \sqrt{x-2}\), the graph begins at the x-coordinate where the domain starts, which is \(x = 2\), and rises slowly to the right.
The plot looks like a half-parabola.
Key points to remember about square root functions include:

• They start from a specific "minimum" point (defined by the domain).
• The graph can only be in the domain's range and cannot have negative values.
• The curve increases as \(x\) increases, showing an upward trend.

Graphing these functions reveals their basic behavior and shows how they fit into complex equations.