Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\frac{|x|}{x}, x=5\)

Short answer

The domain of the function \(f(x)=\frac{|x|}{x}\) is all real numbers except 0. The range of the function is \(-1\) and \(1\). The value of the function at \(x = 5\) is \(1\).

Step-by-step solution

Find the Domain

The domain of a function is the set of all possible input values. In this case, it is important to note that x cannot be zero because this would result in division by zero, which is undefined. Thus, the domain of the function \(f(x)=\frac{|x|}{x}\) is all real numbers except zero.

Determine the Range

The range of a function is the set of all possible output values. For this function, when x is positive, the output is positive 1, whereas when x is negative, the output is negative 1. Hence, the range of \(f(x)=\frac{|x|}{x}\) is \(-1\) and \(1\).

Evaluate the Function

To evaluate the function at \(x = 5\), simply substitute \(5\) for \(x\) in the function, i.e., \(f(5)=\frac{|5|}{5}\). The absolute value of 5 is 5, so \(f(5)\) equals 1.

Key concepts

Absolute Value Function

Understanding the absolute value function is essential because it determines how we treat positive and negative numbers within calculations. The absolute value of a number is the distance between that number and zero on a number line, and is always non-negative. For example, the absolute value of both -3 and 3 is 3. When dealing with absolute value, you can effectively think of "ignoring" the sign of the number.

In the function given, \(f(x)=\frac{|x|}{x}\), the absolute value \(|x|\) ensures that the numerator is always positive, turning any negative input into a positive by virtue of the absolute process. This mechanism makes it important to understand as our function's behavior depends on it significantly.

Here are the key points regarding absolute value:

• It transforms negative numbers into their positive counterparts.
• Positive numbers and zero remain unchanged.
• Crucial in determining how the output of a function changes with different inputs.

Division by Zero

One of the fundamental rules in mathematics is that division by zero is undefined. This concept is crucial in determining the domain of functions since any situation where division by zero could occur results in the function being undefined at that point.

In our given function, \(f(x)=\frac{|x|}{x}\), the denominator has the potential to reach zero, particularly when \(x = 0\). Thus, the domain of this function includes all real numbers except zero in order to avoid an undefined expression. Understanding this concept prevents erroneous calculations and ensures the function's expression stays within valid limits.

Some important notes about division by zero:

• Never attempt to divide by zero, as it leads to undefined results.
• Always check the denominator when working with fractions to determine the valid domain.
• If a function seems undefined at a point, it's often due to this rule.

Function Evaluation

Function evaluation involves substituting a given value into a function to find the corresponding output. This is a basic, yet powerful, mathematical process that gives a specific instance of how the function behaves.

To evaluate \(f(x)=\frac{|x|}{x}\) at \(x=5\), we substitute 5 for \(x\). Since our absolute value function ensures the numerator remains positive, \(f(5)=\frac{|5|}{5}=\frac{5}{5}=1\). This shows how function evaluation works in concert with other mathematical concepts like absolute value.

Some valuable insights into function evaluation include:

• Substitute the given \(x\)-value and solve for the output.
• Ensure you understand all the operations involved (e.g., absolute value).
• Use function evaluation to check specific points of a function's graph or relationship.