Question

Which of the following functions has a graph that is symmetric about the \(y\) -axis? (a) \(y=\sqrt{x}\) (b) \(y=|x|\) (c) \(y=x^{3}\) (d) \(y=\frac{1}{x}\)

Short answer

The function \(y=|x|\) is symmetric about the y-axis.

Step-by-step solution

Understand Symmetry about the y-axis

A function's graph is symmetric about the y-axis if substituting \(x\) with \(-x\) results in the original function. Mathematically, \(f(x) = f(-x)\).

Test each function for symmetry

Apply the definition to each function. For function (a) \(y=\text{sqrt}(x)\), \(y = \text{sqrt}(-x)\). This is invalid for negative values, so it is not symmetric. For function (b) \(y=|x|\), \(y=|-x| = |x|\), which is the same as the original function, meaning it is symmetric.

Continue testing remaining functions

For function (c) \(y=x^3\), \(y = (-x)^3 = -x^3\), which is not the same as the original function, so it is not symmetric. For function (d) \(y=\frac{1}{x}\), \(y = \frac{1}{-x} = -\frac{1}{x}\), which is not the same as the original function, so it is not symmetric.

Conclude with the symmetric function

Among all the tested functions, only \(y=|x|\) is symmetric about the y-axis.

Key concepts

y-axis symmetry

The concept of y-axis symmetry in functions is an essential aspect to understand when analyzing graphs. It means that the graph of the function looks the same on both sides of the y-axis. In simpler terms, if you were to fold the graph along the y-axis, both halves would align perfectly.
For a function to exhibit y-axis symmetry, it must satisfy the property that substituting \(x\) with \(-x\) results in the same function. Mathematically, this is written as \(f(x) = f(-x)\).
To determine if a function has y-axis symmetry, follow these steps:

• Replace \(x\) with \(-x\) in the function.
• If the new function is equivalent to the original, then the function is symmetric about the y-axis.

Let's apply this definition to some examples:

• For \(y = \sqrt{x}\), substituting \(-x\) results in \(\text{\rmsqrt}{-x}\), which is invalid for negative \(x\). Hence, it is not symmetric.
• For \(y = |x|\), substituting \(x\) with \(-x\) results in \(\text{|}-x\text{|} = |x|\). This shows it is symmetric about the y-axis.

even functions

Even functions are closely related to the concept of y-axis symmetry. An even function is one where the function satisfies the condition \(f(x) = f(-x)\) for all \x\ in the domain of the function.
Even functions have graphs that are symmetric about the y-axis. This means that if you reflect the graph across the y-axis, it will look exactly the same.
Some common examples of even functions include:

• Constant functions, such as \(f(x) = c\), where c is a constant.
• The quadratic function, \(f(x) = x^2\).
• The absolute value function, \(f(x) = |x|\).

Remember, not all functions are even. The graphical symmetry about the y-axis is a quick visual indicator, but always check the condition \(f(x) = f(-x)\) algebraically to be certain.

absolute value function

The absolute value function, written as \(f(x) = |x|\), is a key example of an even function. Its graph is symmetric about the y-axis.
Absolute value represents the distance of a number from zero on the number line, regardless of direction. This distance is always a non-negative number.
The graph of \(|x|\) forms a \