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Chapter 2: Problem 40

Exer. 37-46: (a) Sketch the graph of \(f\) (b) Find the domain \(D\) and range \(R\) of \(f\) (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$f(x)=x^{2}-1$$

Short Answer

Expert verified
Domain: all reals; Range: [-1, 鈭�); Decreasing: (-鈭�, 0); Increasing: (0, 鈭�).

Step by step solution

01

Sketch the Graph

The function given is a quadratic function of the form \(f(x) = ax^2 + bx + c\). In this case, \(a=1\), \(b=0\), and \(c=-1\). The graph is a parabola opening upwards because \(a > 0\). The vertex of the parabola (the minimum point) can be found using \(x = \frac{-b}{2a} = 0\). The vertex is therefore at the point \((0, -1)\). Mark this point on a coordinate graph. The parabola is symmetric about the y-axis. Plot additional points such as \((-1, 0)\) and \((1, 0)\) to finalize the graph sketch.
02

Find the Domain and Range

The domain \(D\) of a quadratic function is all real numbers since there are no restrictions on \(x\). This can be written as \(D: (-\infty, \infty)\). For the range \(R\), since the parabola opens upwards and the vertex is the lowest point, the minimum value of \(f\) is \(-1\). Hence, the range is \(R: [-1, \infty)\).
03

Determine Intervals of Increase or Decrease

Since the vertex \((0, -1)\) is the lowest point, the function is decreasing on the interval \((-\infty, 0)\) and increasing on the interval \((0, \infty)\). Therefore, there are no intervals on which \(f\) is constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
Graphing quadratic functions involves making a 'U-shaped' curve known as a parabola. For the function \( f(x) = x^2 - 1 \), this graph is a classic example of a parabola. Here's how you can sketch one:
  • Since the leading coefficient \( a = 1 \), which is greater than zero, the parabola opens upwards.
  • The equation can be written in the standard form \( f(x) = ax^2 + bx + c \), with \( a = 1, b = 0, \) and \( c = -1 \).
  • The vertex, the lowest point on the graph, can be found using the formula \( x = \frac{-b}{2a} \). For our function, this calculation results in \( x = 0 \), so our vertex is at \( (0, -1) \).
  • Because it opens upwards, the parabola is symmetrical about the y-axis.
  • Additional points such as \( (-1, 0) \) and \( (1, 0) \) lie on the parabola, helping to sketch the curve more accurately.
Graphing these points and connecting them smoothly will give a clear picture of the parabola on the coordinate plane.
Domain and Range
When discussing quadratic functions, determining the domain and range helps us understand the extent of the function's influence.
  • The domain of any quadratic function is all real numbers. For \( f(x) = x^2 - 1 \), this translates to \( D: (-\infty, \infty) \).
  • The range, however, is influenced by the direction and position of the parabola. Since our parabola opens upwards, the lowest point is the vertex at \( (0, -1) \). Therefore, the range is \( R: [-1, \infty) \).
  • This indicates that \( f(x) \) can take any value greater than or equal to \(-1\).
Comprehending domain and range ensures we grasp what values \( f(x) \) can assume for any input \( x \).
Increasing and Decreasing Intervals
The behavior of a quadratic function can change as \( x \) moves along the x-axis, altering whether \( f(x) \) is increasing or decreasing.
  • For the function \( f(x) = x^2 - 1 \), the vertex \( (0, -1) \) serves as a pivot point. The function is not constant anywhere.
  • On the interval \( (-\infty, 0) \), the function is decreasing because the graph moves downward as \( x \) increases towards zero.
  • Beyond the vertex, from \( (0, \infty) \), the function is increasing, as the graph moves upward as \( x \) increases.
Understanding these intervals provides insight into how the quadratic function behaves across different sections of the graph. Recognizing where a function increases or decreases supports solving more complex problems in calculus and optimization contexts.

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Most popular questions from this chapter

Find (a) ( \(f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\) $$f(x)=\frac{x+2}{x-1}, \quad \quad \quad \quad \quad \quad g(x)=\frac{x-5}{x+4}$$

Sketch the graph of the equation. $$y=\left|x^{3}-1\right|$$

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