Problem 4 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator. You may have to ignore some false lines on the graph. Graphing in "dot mode" will also eliminate false lines. $$ f(x)=\frac{1}{(x-1)^{2}} ; \text { find } f(-1) $$

Step-by-Step Solution

Answer

f(-1) = 1/4; Domain: (-∞, 1) ∪ (1, ∞); Range: (0, ∞).

Step 1: Evaluate f(-1)

Plug in \(-1\) into the function \(f(x) = \frac{1}{(x-1)^2}\):\[f(-1) = \frac{1}{((-1)-1)^2} = \frac{1}{(-2)^2} = \frac{1}{4}.\] So, \(f(-1) = \frac{1}{4}.\)

Step 2: Find the Domain of f(x)

The function \(f(x) = \frac{1}{(x-1)^2}\) is undefined when the denominator is zero. Solve \((x-1)^2 = 0\) to find the values that make the denominator zero:\[x-1 = 0 \Rightarrow x = 1.\]Hence, the domain of \(f(x)\) is all real numbers except \(x = 1\). In interval notation, this is \((-\infty, 1) \cup (1, \infty).\)

Step 3: Find the Range of f(x)

The function \(f(x) = \frac{1}{(x-1)^2}\) takes positive values only because the square of a number is always non-negative, and the reciprocal of a positive number is positive. As \(x\) approaches 1, \(f(x)\) approaches infinity due to the denominator approaching zero from either side. Therefore, the range of \(f(x)\) is \((0, \infty).\)

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial. It tells us all the possible inputs (x-values) that the function can accept. To find the domain, we look for any restrictions that might prevent certain values from being used. For example, with the function \(f(x) = \frac{1}{(x-1)^2}\), the denominator should not be zero, as division by zero is undefined. Here, solving \((x-1)^2 = 0\) gives us \(x = 1\). Thus, any real number except \(x = 1\) is allowed. Use interval notation for clarity:

• All real numbers except \(x = 1\), or \((-\infty, 1) \cup (1, \infty)\).

Breaking down domains ensures your function works smoothly over allowed x-values.

Range of a Function

The range of a function represents all the potential outputs (y-values). Discovering the range requires understanding how function behaves and what values it can produce. For \(f(x) = \frac{1}{(x-1)^2}\), notice that because squaring any real number yields a non-negative outcome, the reciprocal \(\frac{1}{{(x-1)^2}}\) results in positives only. As \(x\) nears 1, the function's value skyrockets, essentially going to infinity. Therefore, \(f(x)\) can take any positive number:

• The range is \((0, \infty)\).

Defining the range helps predict the outputs based on conceivable inputs, making it easier to understand the function’s behavior.

Evaluating Functions

Evaluating a function means finding the output (y-value) for a particular input (x-value). To do this, substitute the given x-value into the function and solve. Taking the function \(f(x) = \frac{1}{(x-1)^2}\) as an example, and given \(x = -1\), substitute to find:

• \(f(-1) = \frac{1}{((-1)-1)^2} = \frac{1}{(-2)^2} = \frac{1}{4}\).

The output, \(f(-1) = \frac{1}{4}\), indicates how the function operates at a specific point. Regular practice of evaluating functions hones problem-solving skills and understanding of how different inputs affect outputs.