Problem 5 - 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. $$ f(x)=2 $$

Step-by-Step Solution

Answer

The domain is \((-\infty, \infty)\), the y-intercept is at (0,2) and the function is symmetric about the y-axis.

Step 1: Understand the Function

The function given is a constant function, specifically, \(f(x) = 2\). This means that for every value of \(x\), the output \(f(x)\) will continuously be \(2\).

Step 2: Determine the Domain

The domain of a function consists of all the possible input values \(x\). For a constant function like \(f(x) = 2\), any real number can be input, so the domain is all real numbers, expressed as \((-\infty, \infty)\).

Step 3: Identify Intercepts

An intercept is where the graph meets the axes. Since this is a horizontal line \(f(x) = 2\), it intercepts the \(y\)-axis at the point \((0, 2)\). There is no \(x\)-intercept since \(f(x)\) never reaches 0.

Step 4: Test for Symmetry

To test for symmetry, we check: - **Even Symmetry (Y-axis symmetry):** \(f(x) = f(-x)\). Since the output is constant, \(f(-x) = 2 = f(x)\), thus it is even. - **Odd Symmetry (Origin symmetry):** \(-f(x) = f(-x)\). This is not valid as \(-2 eq 2\). Thus, the function is not odd. It is symmetric about the \(y\)-axis.

Step 5: Sketch the Graph

Draw a horizontal line at \(y = 2\) across the width of the graph. Mark the \(y\)-intercept at \((0, 2)\). The graph is a straight, horizontal line, showing it is constant at \(y = 2\).

Key Concepts

Domain of a Function

The domain of a function is an essential component when working with any type of function. Simply put, the domain tells us all the possible values that can be input into the function. For the constant function, like our example given as \(f(x) = 2\), the domain includes all real numbers.

• A constant function like \(f(x) = 2\) remains at the same value irrespective of the input \(x\).
• This means you can input any real number, from negative infinity to positive infinity.
• Thus, we express the domain in interval notation as \((-, )\).

Always remember that for constant functions, the domain is unrestricted since you can choose any real number and still obtain a valid output.

Graph Symmetry

Graph symmetry is a fascinating aspect that reveals much about a function's behavior and simplicity in analysis. For symmetry, we generally consider two types: even (y-axis) and odd (origin) symmetry.

• With constant functions, like \(f(x) = 2\), the function is symmetric about the y-axis.
• This means if you replace \(x\) with \(-x\), you get the same output \(f(x) = 2 = f(-x)\).
• However, it's not symmetric about the origin since \(-f(x)\) doesn't equal \(f(-x)\).

In simpler terms, if you visualize the horizontal line of \(f(x) = 2\), it appears the same when mirrored over the y-axis, confirming its even symmetry.

Intercepts

Intercepts occur where a function crosses the axes. Understanding these points can provide valuable insights into a graph's initial behavior.

• In a constant function such as \(f(x) = 2\), it naturally includes a y-intercept because the line crosses the y-axis at \(y = 2\).
• The specific intercept point on the graph is \((0, 2)\).
• There are no x-intercepts present here because this function's value never drops to zero on the y-axis.

To summarize, for a graph of a constant function, the relevant intercepts are the y-intercepts, and they play a crucial role in defining and plotting the function correctly.