Problem 20 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$g(x)=x^{2}-1$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = x^{2}-1\) is all real numbers, (-\(\infty\), \(\infty\)). The range of the function is [-1, \(\infty\)).

Step 1: Identify the type of function and its domain

The function \(g(x) = x^{2}-1\) is a quadratic function which is a parabola. For this type of function, the domain is all real numbers since x can take any number and can be plugged in without causing any undefined results. Thus the domain is \( \) (-\(\infty\), \(\infty\)) \(-\(\infty\) <= x <= \(\infty\)).

Step 2: Find the minimum or maximum point of the function

For a quadratic function of the form \(f(x) = ax^{2} + bx + c\) its vertex (minima or maxima) is given by the point \(-b/2a, f(-b/2a)\). Here our function is \(g(x) = x^{2}-1\), the coefficient a=1 (coefficient of \(x^{2}\)) and there is no x term, so b=0. Thus the minimum point (as a is positive, so the parabola opens upwards) is \(0, g(0) = 0^{2}-1 = -1\).

Step 3: Identify the range of the function

The range of the function is all possible y-values. As our function is set up, it has a minimum point at y = -1 and will go up to positive infinity as x approaches both positive and negative infinity. So, the range of this function is [-1, \(\infty\)) or -1<= y <\(\infty\).

Key Concepts

Function Domain

The function domain refers to the set of all possible input values (x-values) that a function can accept. For quadratic functions like \(g(x) = x^2 - 1\), the domain is typically all real numbers. This means that you can plug in any real number into the function, and it will yield a valid output.

That's because quadratic functions are polynomials, and polynomials are defined for any real number.

• No matter if \(x\) is positive, negative, or zero, \(x^2\) will always produce a valid output without any restrictions.
• This makes the domain of \(g(x) = x^2 - 1\) the entire set of real numbers, often noted as \((-\infty, \infty)\).

Understanding the domain is crucial because it informs you of all the possible x-values that you can use in your computations without running into undefined expressions. This is especially useful when solving real-world problems.

Function Range

The function range is the set of all possible output values (y-values) that result from using the function. For the quadratic function \(g(x) = x^2 - 1\), the range is determined by the shape and position of the parabola it forms.

Since the quadratic function is in the form of a parabola, we need to find its vertex to determine the lowest or highest point. In our case, the vertex is at the minimum point because the parabola opens upwards.

• The vertex of the parabola \(g(x) = x^2 - 1\) is at the point (0, -1).
• This tells us that the smallest y-value possible is -1.
• As x moves towards positive or negative infinity, the parabola rises towards positive infinity.

Thus, the range of \(g(x) = x^2 - 1\) starts from -1 and goes up to positive infinity, which we express as \([-1, \infty)\). This implies that any y-value greater than or equal to -1 is achievable by using appropriate x-values in the function.

Parabola

A parabola is the curve made by a quadratic function. For the function \(g(x) = x^2 - 1\), the graph is a parabola that opens upwards.

Parabolas have unique properties: they are symmetrical, have a vertex, and either open upwards or downwards.

• The vertex of the parabola \(g(x) = x^2 - 1\) is at the point (0, -1).
• Since the leading coefficient (the number in front of \(x^2\)) is positive, the parabola opens upwards.
• This means that the vertex is the minimum point, making \(-1\) the lowest y-value.

Recognizing the direction and position of the parabola helps understand the behavior of the quadratic function. For example, knowing the vertex and direction tell us about the maximum or minimum values of the function, as well as about its domain and range. Parabolas are a fundamental concept in algebra and help model various physical phenomena, like projectile motion.