Problem 19 - Algebra 2

Question

Sketch the graph of each function. Then state the function's domain and range. $$ y=5(2)^{x} $$

Step-by-Step Solution

Answer

The domain is all real numbers; the range is all positive real numbers.

Step 1: Understand the Function

The given function is an exponential function, expressed as \( y = 5(2)^x \). Such functions exhibit rapid growth or decay depending on the base. Here, the base is \(2\), which is greater than 1, indicating the function represents exponential growth.

Step 2: Determine the Domain

Exponential functions of the form \( y = a(b)^x \) have a domain of all real numbers, \( (-\infty, \infty) \). This means \( x \) can take any real value.

Step 3: Determine the Range

Since \( 5(2)^x \) is an exponentially growing function with all positive outputs, its range is all positive real numbers \( (0, \infty) \). This is because no value of \(2^x\) will yield a negative or zero value when multiplied by 5.

Step 4: Identify Key Points

Select some key values for \( x \) to get corresponding points on the graph. For example: - When \( x = 0 \), \( y = 5(2)^0 = 5 \)- When \( x = 1 \), \( y = 5(2)^1 = 10 \)- When \( x = -1 \), \( y = 5(2)^{-1} = 2.5 \)- When \( x = 2 \), \( y = 5(2)^2 = 20 \).These points will help plot the curve accurately.

Step 5: Sketch the Graph

Use the key points from Step 4 to sketch the graph. Plot the points \((0, 5)\), \((1, 10)\), \((-1, 2.5)\), and \((2, 20)\). Connect these points with a smooth curve, keeping in mind the graph will continuously rise as \( x \) increases and approach zero but never touch it as \( x \) decreases.

Step 6: State the Domain and Range

Based on the analysis:- The domain of \( y = 5(2)^x \) is \( (-\infty, \infty) \).- The range of \( y = 5(2)^x \) is \( (0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented as \( x \)) that will yield a valid output \( y \). For exponential functions, particularly those of the form \( y = a(b)^x \), the domain is often straightforward. As seen in the function \( y = 5(2)^x \), the domain encompasses all real numbers, denoted as \( (-\infty, \infty) \). This is because you can insert any real number for \( x \) without encountering mathematical anomalies that would invalidate the function, like division by zero or square roots of negative numbers.

To reiterate:

• Exponential functions like \( y = 5(2)^x \) can accommodate any real number for \( x \).
• The domain is unbounded, meaning it stretches infinitely in both the positive and negative directions.

Understanding that the domain is all real numbers gives freedom in picking \( x \) values when analyzing or graphing the function.

Range of a Function

While the domain provides insight into potential inputs, the range speaks to what outputs a function can produce. For the exponential function \( y = 5(2)^x \), the range is exclusively positive real numbers, expressed as \( (0, \infty) \). This is due to the property of exponential growth, where the base, \( 2 \), is positive and greater than 1. Consequently, \( (2)^x \) is always positive, and when multiplied by the positive constant 5, it only yields positive results.

Key points to remember:

• The range begins just above zero and ascends infinitely.
• Outputs of \( y \) are never zero or negative for exponential growth functions like \( y = 5(2)^x \).

This consistent positivity of \( y \) values characterizes many exponential functions and is crucial for defining their behavior.

Graphing Functions

Graphing functions provides a visual representation of their behavior over various inputs. For \( y = 5(2)^x \), plotting the graph involves finding key points through selected \( x \) values and corresponding \( y \) calculations. As exemplified in the step-by-step solution, we choose values like \( x = 0, 1, -1, \) and \( 2 \) to plot points including \((0, 5)\), \((1, 10)\), \((-1, 2.5)\), and \((2, 20)\). Connecting these points with a smooth curve helps sketch the exponential growth.

Graphing steps:

• Choose a variety of \( x \) values, both negative, zero, and positive.
• Calculate \( y \) for each \( x \).
• Plot the resulting points with careful attention to their position.
• Draw a smooth curve through the points to approximate the function's continuous nature.

Features of exponential functions to observe include their upward direction for growth and the tendency to approach but never reach zero along the \( x \)-axis as \( x \) decreases. This provides a telling depiction of exponential behavior, aiding comprehension.