Problem 15 - Precalculus : Building Concepts and Connections

Question

This set of exercises will reinforce the skills illustrated in this section. Graph each pair of functions on the same set of coordinate axes, and find the domain and range of each function. $$f(x)=(x+1)^{2}, \quad g(x)=x^{2}$$

Step-by-Step Solution

Answer

For both the functions \(f(x) = (x+1)^2\) and \(g(x) = x^2\), the domain is all real numbers, and the range is non-negative real numbers, \(y \ge 0\).

Step 1: Graphing the function \(f(x)\)

In order to graph the function \(f(x) = (x+1)^2\), you can use any graphing tool. For manual plotting, remember that \(f(x) = (x+1)^2\) is a vertical shift of the function \(f(x) = x^2\) by 1 unit to the left. Thus, the vertex of this parabola is at \((-1, 0)\), and the graph opens upward.

Step 2: Graphing the function \(g(x)\)

To graph the function \(g(x) = x^2\), we still remember that this is a basic quadratic function with a vertex at the origin \((0, 0)\) and it also opens upward.

Step 3: Finding the domain and range of the function \(f(x)\)

For the function \(f(x) = (x+1)^2\), the domain is all real numbers because we can substitute any real number for \(x\) to get a valid output. The range is all non-negative real numbers because when squaring any real number, the output will be either 0 or positive. Thus, Domain: \(x \in R\) and Range: \(y \ge 0\).

Step 4: Finding the domain and range of the function \(g(x)\)

For the function \(g(x) = x^2\), the domain is once again all real numbers. The range is also non-negative real numbers. Thus, Domain: \(x \in R\) and Range: \(y \ge 0\)

Key Concepts

Domain and Range of a Function

Understanding the domain and range of a function is crucial when graphing quadratic functions like f(x) = (x+1)^2 and g(x) = x^2. The domain of a function refers to the complete set of possible values of the independent variable. In simpler terms, it's the set of all 'x' values that you can input into the function.

For both functions f(x) and g(x), the domain includes all real numbers, denoted as \( x \in \mathbb{R} \). In practice, this means any real number can be squared, even if it’s negative, zero, or positive. Regarding the range, which represents the possible output or 'y' values of the function, these particular quadratic functions will only yield results that are zero or positive. This is because the square of any real number cannot be negative. Consequently, our range is \( y \geq 0 \), indicating that the vertex of the parabola is the minimum point and the graph extends infinitely upwards.

Quadratic Functions Shifts

When graphing quadratic functions, it is important to recognize how shifts transform the graph. A quadratic function in standard form is y = x^2. Any changes to this format can shift the graph horizontally or vertically.

For the function f(x) = (x+1)^2, we notice a horizontal shift when compared to g(x) = x^2. In f(x), the '(+1)' within the parentheses shifts the graph 1 unit to the left. This is because the function is effectively squared after decreasing 'x' by 1, hence the leftward movement on the graph. This kind of shift does not affect the domain or range but changes the position of the vertex, which in the case of f(x), moves from \( (0, 0) \) to \( (-1, 0) \).

Understanding these shifts is valuable when graphing as they dictate the starting point of the parabola on the coordinate axes, and once the vertex is located, the rest of the graph can be shaped accordingly.

Vertex of a Parabola

The vertex of a parabola is a significant feature in its graph. For the quadratic functions we are considering, the vertex represents the lowest point on the graph for functions that open upwards, like f(x) = (x+1)^2 and g(x) = x^2. The coordinates of the vertex are found at the turning point where the parabola changes direction.

For the basic quadratic function g(x) = x^2, the vertex is at the origin, \( (0, 0) \). For f(x) = (x+1)^2, which is the function g(x) shifted to the left by 1 unit, the vertex is at \( (-1, 0)\). The 'x' coordinate of the vertex for any quadratic function in the form y = (x - h)^2 is given by 'h'. The 'y' coordinate is the minimum value of the function if it opens upwards, which can be determined by substituting 'h' back into the function.

Knowing the vertex helps in graphing the entire parabola since it acts as the fundamental reference point, around which the `U` shape of the graph is symmetrical.