Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=(x-1)^{2}-5$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \([-5, \infty)\)

Step 1: Identify the Basic Function

The given function is \( f(x)=(x-1)^{2}-5 \). Notice that it is based on the basic quadratic function \( g(x) = x^2 \), which is a parabola opening upwards.

Step 2: Determine the Domain

For any quadratic function, including \( (x-1)^2 \), the domain is all real numbers because there is no restriction on \( x \). Since \( (x-1)^2 \) is defined for every real number, so is \( f(x)=(x-1)^2-5 \). Therefore, the domain is \( (-\infty, \infty) \).

Step 3: Determine the Range

To find the range, analyze the transformed function. The basic function \( x^2 \) has a minimum value of 0. The transformation \( (x-1)^2 \) shifts the parabola right by 1 unit, while \( -5 \) shifts it down by 5 units. Thus, the minimum value of the function \( f(x) \) is \( 0 - 5 = -5 \). Therefore, the range is \( [-5, \infty) \).

Step 4: Conclusion

We've identified both the domain and the range for the function \( f(x) = (x-1)^2 - 5 \). Verify by considering the impact of transformations on the parabolic graph.

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of possible input values (usually represented as \( x \)) that satisfy the function's structure. For quadratic functions like \( f(x) = (x-1)^2 - 5 \), the domain is quite straightforward. Quadratic functions are polynomial functions and inherently continuous over the real numbers.

• Which means: there are no breaks, gaps, or restrictions.
• The expression \((x-1)^2\) covers every real number \( x \) can take, from \(-\infty\) to \(+\infty\).

This universality stems from the fact that squaring any real number results in a valid output. Thus, the domain of \( f(x) = (x-1)^2 - 5 \) includes all real numbers.
To express this formally, we denote the domain by the interval \((-\infty, \infty)\). This indicates that there isn't a single real number that cannot be squared and then shifted as described in the function.

Range of a Function

Unlike the domain, the range of a function specifies all possible output values (represented typically as \( y \) or \( f(x) \)). When dealing with a quadratic function, finding the range involves examining its transformation from its parent function, usually \( y = x^2 \).

• The standard parabola \( y = x^2 \) has its lowest point at \( y = 0 \).
• With transformations, \( f(x) = (x-1)^2 - 5 \) sees the parabola shifted downward by 5 units.

This applies when you consider that \( y \) can reach a minimum value of \(-5\) at its vertex.
Thus, this transformation results in the new range \( y \geq -5 \), meaning the output values start from \(-5\) and extend infinitely upwards. This can be recorded using interval notation as \([-5, \infty)\).

Graph Transformations

Graph transformations adjust a basic function graph according to prescribed changes in the equation. It's insightful to visualize these transformations to understand quadratic functions like \( f(x) = (x-1)^2 - 5 \).

• Horizontal Shifts: The \((x-1)^2\) changes the position of the parabola, moving it one unit right.
• Vertical Shifts: The \(-5\) at the end diminishes all output values by 5 units, thus lowering the whole graph.

The vertex, initially at the origin in \( y = x^2 \), moves to \( (1, -5) \) due to these transformations.
Understanding how graph transformations work helps us visually plot the graph without drawing every detail manually. Each transformation directly correlates to how we calculate domain and range for quadratic functions, anchoring critical concepts such as minimum or maximum values of a function.