Textbook Solutions

Explanation:

For two functions to be considered equal, they must satisfy two conditions:

1. They must have the exact same domain.

2. For every value 'x' in their domain, their outputs must be the same (i.e., f(x) = g(x)).

Let's examine the two functions based on these conditions.

1. Determine the domains:

• Domain of f(x): The function f(x) = (x^2 - x) / (x - 1) is a rational function. Its domain includes all real numbers except for values that make the denominator zero.
  To find the excluded value, we set the denominator to zero:
  x - 1 = 0
  x = 1
  So, the domain of f(x) is all real numbers except for x = 1.

• Domain of g(x): The function g(x) = x is a simple linear function. Its domain is all real numbers, with no restrictions.

Comparison:

The domain of f(x) is {x | x is a real number and x ≠ 1}.
The domain of g(x) is {x | x is a real number}.

Since the domains are not the same, the functions f and g are not equal. The first condition for equality fails.

Further Analysis:

Although the domains are different, let's simplify the expression for f(x). For any x ≠ 1, we can do the following:

f(x) = (x^2 - x) / (x - 1)
f(x) = x(x - 1) / (x - 1)
f(x) = x

This shows that f(x) = g(x) for all values of x except at x = 1. However, at x = 1:

• f(1) is undefined (because it causes division by zero).

• g(1) = 1.

Because f(x) and g(x) do not have the same domain, they are not the same function. Graphically, g(x) is a continuous straight line, while f(x) is the same line but with a "hole" or point of discontinuity at x = 1.

(a) State the values of f(-4) and g(3).
To find f(-4), locate x = -4 on the horizontal axis and find the corresponding y-value on the blue curve (f).

• f(-4) = 2

To find g(3), locate x = 3 on the horizontal axis and find the corresponding y-value on the red curve (g).

• g(3) = -1

(b) Which is larger, f(-3) or g(-3)?
First, find the values of f(-3) and g(-3) from the graph.

• At x = -3, the blue curve f is at a y-value of approximately 1.5. So, f(-3) ≈ 1.5.

• At x = -3, the red curve g is at a y-value of -2. So, g(-3) = -2.
  Since 1.5 is larger than -2, f(-3) is larger than g(-3).

(c) For what values of x is f(x) = g(x)?
This occurs where the two graphs intersect.
The graphs intersect at two points. By reading their x-coordinates:

• x = -2 and x = 2

(d) On what interval(s) is f(x) ≤ g(x)?
This is the interval where the blue curve (f) is below or at the same height as the red curve (g).
Looking at the graph, this happens between the two intersection points (inclusive).

• The interval is [-2, 2].

(e) State the solution of the equation f(x) = -1.
We need to find the x-value where the blue curve (f) has a y-value of -1.
The entire blue curve is above the x-axis, meaning f(x) is always positive. Its minimum value is greater than 0. Therefore, f(x) can never be equal to -1.

• There is no solution.

(f) On what interval(s) is g decreasing?
A function is decreasing where its graph slopes downward from left to right.
The red curve (g) reaches its maximum point at x=0 and then decreases until its endpoint at x=4.

• The interval is [0, 4].

(g) State the domain and range of f.

• Domain: The set of all x-values for the blue curve. The graph extends from x = -5 to x = 3.
  Domain of f: [-5, 3]

• Range: The set of all y-values for the blue curve. The lowest point is at x = -1, where the y-value is approximately 0.5. The highest point is at the endpoints, where y = 3.
  Range of f: [0.5, 3] (approximately)

(h) State the domain and range of g.

• Domain: The set of all x-values for the red curve. The graph extends from x = -4 to x = 4.
  Domain of g: [-4, 4]

• Range: The set of all y-values for the red curve. The lowest point is at x = -4, where y = -4. The highest point is at x = 0, where y = 3.
  Range of g: [-4, 3]

Example 1: Total Cost as a Function of an Item's Weight

• Verbal Description: The total cost of buying a specific type of produce, like apples or bananas, is a function of the weight of the produce you buy.

• Domain and Range:

  • Domain (Input): The weight of the produce. The domain would be all positive real numbers, starting from 0. You can't buy negative weight. A practical upper limit might exist (e.g., how much the store has in stock), but theoretically, it's [0, ∞).

  • Range (Output): The total cost. The range would also be all positive real numbers, starting from $0. If you buy 0 pounds, the cost is $0. The cost increases as the weight increases. The range is [0, ∞).

• Rough Graph Sketch:

  • The graph would be a straight line starting from the origin (0, 0).

  • The x-axis represents the weight, and the y-axis represents the cost.

  • As you move to the right (increasing weight), the line goes up at a constant rate.

  • The steepness (slope) of the line would be the price per unit of weight (e.g., dollars per pound).

Example 2: A Car's Remaining Fuel as a Function of Distance Driven

• Verbal Description: The amount of fuel remaining in a car's tank is a function of the distance driven since the last time the tank was filled.

• Domain and Range:

  • Domain (Input): The distance driven. The domain starts at 0 (immediately after filling up). It ends when the car runs out of gas. For a car with a 15-gallon tank that gets 30 miles per gallon, the maximum distance is 450 miles. So, a practical domain would be [0, 450].

  • Range (Output): The amount of fuel in the tank. The range starts at the maximum capacity of the tank (e.g., 15 gallons) and ends at 0. So, the range would be [0, 15].

• Rough Graph Sketch:

  • The graph would be a straight line with a negative slope.

  • The x-axis represents the distance driven, and the y-axis represents the fuel remaining.

  • It starts at its highest point on the y-axis (a full tank) when x=0.

  • As you move to the right (increasing distance), the line goes down until it hits the x-axis (empty tank).

Example 3: A Person's Height as a Function of Their Age

• Verbal Description: The height of a particular person is a function of their age.

• Domain and Range:

  • Domain (Input): The person's age. The domain starts at 0 (birth) and ends at their age of death. A typical human lifespan could be represented by the interval [0, 90].

  • Range (Output): The person's height. The range starts at their height at birth (e.g., 20 inches) and goes up to their maximum adult height (e.g., 70 inches). The range would be approximately [20, 70].

• Rough Graph Sketch:

  • This graph would not be a straight line.

  • The x-axis represents age, and the y-axis represents height.

  • The graph starts at a positive value on the y-axis (birth height).

  • It rises steeply during the first few years of life (rapid growth in childhood).

  • The curve becomes less steep during teenage years.

  • It then becomes flat (horizontal) for most of adult life, as the person's height remains constant.

  • It might even curve slightly downward in old age as people can lose a small amount of height.

To determine if an equation defines y as a function of x, we need to check if for every possible input value of x, there is exactly one corresponding output value of y. The easiest way to do this is to solve the equation for y.

1. Start with the given equation:
   3x² - 2y = 5

2. Isolate the term containing y. To do this, subtract 3x² from both sides of the equation:
   -2y = 5 - 3x²

3. Solve for y. Divide both sides by -2:
   y = (5 - 3x²) / -2

4. Simplify the expression:
   y = -5/2 + (3/2)x²
   or
   y = (3/2)x² - 5/2

Conclusion:

The resulting equation, y = (3/2)x² - 5/2, shows that for any value we substitute for x, there is only one possible calculation to find the value of y. Since each input x yields exactly one output y, the equation does define y as a function of x.