Problem 48 - Algebra 1: Concepts and Skills

Question

Using your graphs , describe the domain and the range of the function. $$y=-5\left(\frac{1}{5}\right)^{x}$$

Step-by-Step Solution

Answer

The domain of the function \(y=-5\left(\frac{1}{5}\right)^{x}\) is all real numbers, represented as \((-∞,∞)\) in interval notation. The range of the function is \((-\infty,0)\).

Step 1: Understand the function format

Recognize that the function is in the form of an exponential decay function \(y = ab^x\), where in this case, \(a = -5\) and \(b = \frac{1}{5}\). This type of function decreases rapidly and then gradually levels off as \(x\) becomes large.

Step 2: Graph the Function

Plot the function on a graph. The y-intercept is at the point (0,-5). As you move to the right in the positive x direction, due as a consequence of b value being \(\frac{1}{5}\), the function decreases rapidly and then gradually levels off approaching y = 0. As you move to the left, the value of the function decreases without limit.

Step 3: Identify the Domain

The domain of a function is the set of all possible x-values. For this function, we can see from the graph that it's possible to substitute any real number for x, so the domain is all real numbers, which in interval notation is \((-∞,∞)\).

Step 4: Identify the Range

The range of a function is the set of all possible y-values. Here, as x becomes increasingly larger, the function value approaches 0 but never actually equals 0. Also, as you move to the left, the function decreases without limit due to the '-' sign. Thus, the range of this function is \((-\infty,0)\).

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of possible values of the independent variable, usually represented as "x". In simpler terms, it's all the values that you can feed into the function without causing any breaks or issues.
For the exponential decay function given as \(y = -5\left(\frac{1}{5}\right)^{x}\), there are no restrictions on what \(x\) could be. This means \(x\) can be any real number.
This is because exponential functions, whether they grow or decay, are continuous and defined for all real \(x\) values. In interval notation, we describe this as \((-∞,∞)\).

When thinking about the domain, always ask yourself:

• Could this \(x\) value cause the function to be undefined? (e.g., dividing by zero or taking the square root of a negative number for real functions)
• Is there any restriction given by the problem itself?

Since there are no such barriers here, the domain stretches infinitely in both directions, encapsulating all real numbers.

Range of a Function

The range of a function is the set of possible outcomes, or 'y-values', that the function can produce. Knowing this helps us understand the behavior and limitations of the function in question.
The function \(y = -5\left(\frac{1}{5}\right)^{x}\) describes exponential decay. As \(x\) increases, the value \(\left(\frac{1}{5}\right)^{x}\) shrinks toward zero. But, because it's multiplied by \(-5\), the entire function approaches zero from the negative side.
So, no matter how large \(x\) gets, \(y\) never actually reaches zero. Instead, it hovers closer and closer, always staying slightly below it. As \(x\) decreases, the value of \(y\) becomes more negatively immense.
Thus, the range of this function, which expresses all the possible outputs, ends up being \((-\infty, 0)\). It's crucial for students to remember that the range reflects where the y-values lie on the graph and is determined by both the direction and the behavior of the exponential decay.

Exponential Decay

Exponential decay describes a process of reduction that decreases rapidly at first and slows over time. It's characterized by the base of an exponential function being a fraction between 0 and 1.
In the function \(y = -5\left(\frac{1}{5}\right)^{x}\), you see exponential decay through the behavior of the base \(\frac{1}{5}\). This means that for every increase in \(x\), the factor by which \(y\) is multiplied gets smaller, hence the 'decay'.
Graphically, these functions slope downwards, starting steep and flattening as \(x\) progresses. This makes exponential decay ideal for modelling real-world situations like cooling, depreciation of assets, or radioactive decay, where the quantity reduces and tapers off over time.

Understanding exponential decay involves noting:

• The initial value or the y-intercept is given by \(a\) (here, -5), which factored by a negative sign inverts the direction of decay.
• The rate of decay hinges on the base, \(\frac{1}{5}\), determining how rapidly the function decreases.

Through these concepts, you can better visualize and understand the implications of exponential decay, linking the mathematics to recognizable real-world behaviors.