a. Plot the data for the functions [tex]V(f(x) V)[/tex] and [tex]V(g(x) V)[/tex] on a grid.
b. Identify each function as linear, quadratic, or exponential, and use complete sentences to explain your choices.
c. Describe what happens to the function values in each function as [tex]V(X)[/tex] increases from left to right.
d. At what value(s) of [tex]V(x)[/tex] are the function values equal? If you cannot give exact values for [tex]V(x)[/tex], give estimates.
Answer (1)
V ( f ( x )) is an exponential function: V ( f ( x )) = 16 1 ⋅ 4 x .
V ( g ( x )) is a linear function: V ( g ( x )) = x .
As x increases, both V ( f ( x )) and V ( g ( x )) increase.
The function values are equal at x ≈ 0.0687 and x ≈ 2.7225 .
Explanation
Understanding the Problem We are given data for two functions, V ( f ( x )) and V ( g ( x )) , and we need to analyze their properties. The data is presented in a table, and we are asked to identify the type of each function (linear, quadratic, or exponential), describe their behavior, and find the values of x where the function values are equal.
Analyzing V(f(x)) First, let's analyze the function V ( f ( x )) . The given data points are ( − 2 , 16 1 ) , ( − 1 , 4 1 ) , ( 0 , 1 ) , ( 1 , 4 ) , and ( 2 , 16 ) . To determine the type of function, we can check the ratios of consecutive y-values:
16 1 4 1 = 4 4 1 1 = 4 1 4 = 4 4 16 = 4
Since the ratio between consecutive y-values is constant (equal to 4), the function V ( f ( x )) is an exponential function. Specifically, it can be represented as V ( f ( x )) = 4 x /16 = 16 1 ⋅ 4 x .
Analyzing V(g(x)) Now, let's analyze the function V ( g ( x )) . The given data points are ( 3 , 3 ) , ( 4 , 4 ) , ( 5 , 5 ) , ( 6 , 6 ) , and ( 7 , 7 ) . To determine the type of function, we can check the differences between consecutive y-values:
4 − 3 = 1 5 − 4 = 1 6 − 5 = 1 7 − 6 = 1
Since the difference between consecutive y-values is constant (equal to 1), the function V ( g ( x )) is a linear function. Specifically, it can be represented as V ( g ( x )) = x .
Behavior of V(f(x)) As x increases from left to right, the function values of V ( f ( x )) increase. This is because the base of the exponential function (4) is greater than 1, so the function grows as x increases. Specifically, as x goes from -2 to 2, V ( f ( x )) goes from 16 1 to 16.
Behavior of V(g(x)) As x increases from left to right, the function values of V ( g ( x )) also increase. This is because the slope of the linear function is positive (equal to 1), so the function grows as x increases. Specifically, as x goes from 3 to 7, V ( g ( x )) goes from 3 to 7.
Finding Where Function Values are Equal To find the values of x where the function values are equal, we need to solve the equation V ( f ( x )) = V ( g ( x )) , which is 16 1 ⋅ 4 x = x . This equation is difficult to solve analytically, so we can estimate the solutions graphically or numerically. Using a numerical solver, we find that the solutions are approximately x ≈ 0.0687 and x ≈ 2.7225 .
Final Answer Therefore, the function V ( f ( x )) is exponential, and the function V ( g ( x )) is linear. As x increases, both function values increase. The function values are equal at approximately x ≈ 0.0687 and x ≈ 2.7225 .
Examples
Understanding different types of functions like linear and exponential functions is crucial in various real-world applications. For instance, population growth can often be modeled using exponential functions, while the cost of producing items might be modeled using a linear function. By analyzing these functions, we can make predictions about future trends and optimize decision-making. For example, a business might use these models to forecast sales or determine the most efficient production level. The points where these functions intersect can represent break-even points or optimal operating conditions, providing valuable insights for strategic planning. Analyzing these functions helps us understand the relationships between different variables and make informed decisions in various fields.
