Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The graph of \(y=f(x)\) is symmetric about the \(y\) -axis. Which of the following functions is equal to \(f(x)\) ? (a) \(g(x)=-f(x)\) (b) \(h(x)=f(-x)\) (c) \(j(x)=-f(-x)\)

Step-by-Step Solution

Answer

The function equal to \( f(x) \) is (b) \( h(x)=f(-x) \)

Step 1: Evaluate Option (a)

In option (a), we have \( g(x)=-f(x) \). Let’s check its symmetry by replacing every \( x \) with \( -x \). We get \( g(-x)=-f(-x) \). This tells us that the function is symmetric about the origin, not the y-axis.

Step 2: Evaluate Option (b)

In option (b), \( h(x)=f(-x) \). Replacing every \( x \) with \( -x \) gives us \( h(-x)=f(-(-x))=f(x) \) which is equal to the original function. This tells us that the function is symmetric about the y-axis, so this is the correct option.

Step 3: Evaluate Option (c) for Completeness

Finally, we evaluate option (c), \( j(x)=-f(-x) \). By replacing every \( x \) with \( -x \) we get \( j(-x)=-f(-(-x))=-f(x) \). This tells us \( j(x) \) is symmetric about the origin, not the y-axis.

Key Concepts

even and odd functions

Understanding even and odd functions can really help when analysing the symmetry in function graphs. **Even functions** have a unique property: their graphs are symmetric around the y-axis. This means if you pick any point on the graph and draw a line to the y-axis, you will find an equivalent point on the opposite side of the axis. This happens because for even functions, the equation satisfies that \[ f(x) = f(-x) \].
On the other hand, **odd functions** have a symmetry around the origin. This symmetry implies that if you rotate the entire graph by 180 degrees around the origin, it would look the same. The mathematical condition for an odd function is \[ f(-x) = -f(x) \].
Identifying the type of symmetry a function has can be a powerful tool when solving problems, like in the exercise provided, where identifying the symmetry helped confirm that option (b) was the correct choice.

function transformations

Function transformations include translations, reflections, stretches, and compressions, which modify how a graph looks without altering its core properties.
One of the most common transformations is a reflection. When reflecting a function across the y-axis, like in option (b) of our exercise, the transformation involves replacing every instance of \( x \) with \( -x \). The function becomes \( f(-x) \), showcasing the y-axis symmetry typical of even functions.
Other types of function transformations include:

• **Translation**: Shifting the graph horizontally or vertically without changing its shape. E.g., \( f(x) + c \) shifts the graph up by \( c \).
• **Vertical stretch or compression**: Multiplying the function by a factor greater than 1 stretches it, and less than 1 compresses it.
• **Horizontal stretch or compression**: Achieved by replacing \( x \) with \( cx \), where stiffness changes based on the value of \( c \).

Recognizing these transformations is critical for correctly interpreting function graphs, especially in distinguishing between symmetric characteristics.

graphical symmetry

Symmetry in math often refers to how one side of an object or equation mirrors the other. In the realm of graphing functions, this concept plays a crucial role in understanding and predicting function behaviors.
**Graphical symmetry** can be of multiple types, but the focus often is on symmetry about the y-axis or the origin. Here, if a graph mirrors across the y-axis, it's said to have y-axis symmetry, indicating it's an even function. Conversely, symmetry about the origin indicates an odd function.
Recognizing these symmetries in graphs can simplify both graph sketching and function analysis. Another aspect to consider is the real-world applications of these symmetrical properties. They are evident in physics, where certain parameters respect symmetry laws, helping scientists make predictions about the universe's workings.
In conclusion, identifying and understanding these types of symmetrical properties in functions allow us to simplify how we solve function-related problems, serving as fundamental tools in mathematics.