Question

In Exercises 49 and \(50,\) (a) draw the graph of the function. Then find its (b) domain and (c) range. $$f(x)=2|x+4|-3$$

Short answer

The graph of the function \(f(x) = 2|x + 4| - 3\) is a V-shape with vertex at (-4,-3). The domain is all real numbers, \((- \infty, \infty)\), and the range is every number greater than or equal to -3, \([-3, \infty)\).

Step-by-step solution

Draw the Graph

The base function here is the absolute value function, which is a V-shaped graph. It is first translated 4 units to the left by the '+4' inside the absolute value. Then, it is scaled vertically by a factor of 2 due to the '2|...|' part. Finally, it is translated 3 units downwards due to the '-3' outside the absolute value. Thus, the vertex of the function would be at (-4,-3), and it would open upwards since the coefficient of the absolute value is positive.

Find Domain

The domain of a function is the set of all possible input values. For this function, there are no restrictions on x since any real number x can be plugged into the function without causing any mathematical issues such as division by zero or taking square roots of negative numbers. Hence the domain is all real numbers, which can be represented in interval notation as \((- \infty, \infty)\).

Find Range

The range of a function is the set of all possible output values. Since this is an absolute value function, it has a minimum value at its vertex, and it will increase without bound as x moves away from -4. The minimum y-value is -3, which is the y-coordinate of the vertex. Thus the range is every number greater than or equal to -3, which can be represented in interval notation as \([-3, \infty)\).

Key concepts

Graphing Functions

Graphing functions is an essential skill in mathematics, allowing us to visually represent the relationship between variables. To graph the given absolute value function, f(x) = 2|x+4| - 3, we start with the basic 'V' shape of the absolute value graph. This 'V' is then modified according to the function's specific transformations.

Understanding each element influencing the graph is crucial. The '+4' inside the absolute value shifts the graph 4 units to the left, changing the vertex from (0, 0) to (-4, 0). Multiplying the absolute value by 2 stretches the graph vertically, doubling the rate at which it increases or decreases from its vertex. Finally, the '-3' outside of the absolute value moves the entire graph down by 3 units, placing the vertex at (-4, -3).

By plotting this new vertex and considering the transforms, a scaled and shifted 'V' shape emerges on your coordinate plane. The graph opens upwards, maintaining the positive slope away from the vertex, indicating that for increasing values of |x+4|, the function's output value will also increase.

Function Domain

The domain of a function refers to all of the input values (x-values) that the function can accept without causing mathematical errors. For the function f(x), since it is based on the absolute value, there are no restrictions stemming from the inner workings of the operation. An absolute value can take any real number and return a non-negative result. Therefore, the domain for our function, f(x) = 2|x+4| - 3, includes all real numbers.

In interval notation, we express this unlimited domain as (-∞, ∞). While the function's transformations move its graph around the coordinate plane, they don't affect the function's ability to take any real number as input. Remembering that the domain includes all possible input values is vital in understanding a function's behavior across the entire spectrum of numbers.

Function Range

In contrast to the domain, the range of a function consists of all the possible output values (y-values) the function can produce. With absolute value functions like f(x) = 2|x+4| - 3, the range is influenced by the nature of the absolute value to produce only non-negative results when unaffected by other transformations.

However, once we consider our function's vertical stretch and downward shift, the minimum output value becomes the y-coordinate of the vertex, which is -3 in this case. As x moves away from -4, the output values increase indefinitely. Hence, the range is every number greater than or equal to -3, and in interval notation, this is written as [-3, ∞). It's crucial to find the y-value of the vertex and understand the graph's direction to determine the range correctly.