Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 19 through 21 (a) find the value of the function at \(x=0, x=1\), and \(x=-1\). (b) find all \(x\) such that the value of the function is \((i) 0,(i i) 1\), and \((i i i)-1 .\) $$ f(x)=\frac{3 x+5}{2} $$

Step-by-Step Solution

Answer

For x=0 the function value is 2.5, for x=1 it's 4, and for x=-1 it's 1. The function reaches the value 0 when x=-5/3, it reaches 1 when x=-3/2, and it reaches -1 when x=-9/2.

Step 1: Evaluate the function at specified points

Substitute the given x-values (0, 1, -1) into the function \(f(x)=\frac{3x+5}{2}\) to find the function values.

Step 2: Calculate function values

For \(x=0\), \(f(0)=\frac{3(0)+5}{2}=\frac{5}{2}\). For \(x=1\), \(f(1)=\frac{3(1)+5}{2}=\frac{8}{2}=4\). For \(x=-1\), \(f(-1)=\frac{3(-1)+5}{2}=\frac{2}{2}=1\).

Step 3: Determine x-values for given function values

To find the x-values for which \(f(x)=0\), \(f(x)=1\), and \(f(x)=-1\), set the function equal to each of these values and solve for x respectively.

Step 4: Solve the equations

For \(f(x)=0\), \(0=\frac{3x+5}{2}\), we get \(x=-\frac{5}{3}\). For \(f(x)=1\), \(1=\frac{3x+5}{2}\), we get \(x=-\frac{3}{2}\). For \(f(x)=-1\), \(-1=\frac{3x+5}{2}\), we get \(x=-\frac{9}{2}\).

Key Concepts

Function Evaluation

Function evaluation is a process where we find the value of a function for specific values of its variable, in this case, the variable is \(x\). In the given problem, the function is \(f(x) = \frac{3x + 5}{2}\). By substituting values \(x = 0\), \(x = 1\), and \(x = -1\) into the function, we can calculate the corresponding outputs.

• For \(x = 0\), substitute 0 into the equation: \(f(0) = \frac{3(0) + 5}{2} = \frac{5}{2}\).
• For \(x = 1\), substitute 1: \(f(1) = \frac{3(1) + 5}{2} = \frac{8}{2} = 4\).
• For \(x = -1\), substitute -1: \(f(-1) = \frac{3(-1) + 5}{2} = \frac{2}{2} = 1\).

Function evaluation helps us understand how the function behaves at different points. It is fundamental in analyzing functions and predicting outcomes.

Linear Functions

Linear functions are mathematical functions that create a straight line when graphed. They can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The function in this problem \(f(x) = \frac{3x + 5}{2}\) is indeed a linear function.

Let's simplify it to see its linear form more clearly. We can rewrite the function as:
\[ f(x) = \frac{3}{2}x + \frac{5}{2} \]
Here, \(\frac{3}{2}\) is the slope and \(\frac{5}{2}\) is the y-intercept. This slope tells us how steep the line is and in which direction it moves. In our function, a positive slope means the line rises as \(x\) increases. Linear functions are very common in mathematical modeling due to their straightforwardness.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. In this exercise, we solve for \(x\) when \(f(x)\) is equal to given values: 0, 1, and -1.

• For \(f(x) = 0\): Write \(0 = \frac{3x + 5}{2}\). Solving for \(x\), we get \(x = -\frac{5}{3}\).
• For \(f(x) = 1\): Use \(1 = \frac{3x + 5}{2}\). Solving gives \(x = -\frac{3}{2}\).
• For \(f(x) = -1\): Set \(-1 = \frac{3x + 5}{2}\). Solving this results in \(x = -\frac{9}{2}\).

Through solving these equations, we identify specific \(x\) values that satisfy each condition, thus enhancing our understanding of how the function behaves at these points.

Mathematical Problem Solving

Mathematical problem solving involves using processes and strategies to find solutions to mathematical questions. In the context of this exercise, it requires understanding the problem, planning a solution strategy, executing the calculations, and reviewing the results.

To address this exercise:

• Understand the function and what is required by reading the problem statement thoroughly.
• Evaluate and compare the function at given \(x\) values.
• Solve for \(x\) where the function equals specific values.
• Review the outcomes to ensure they are logical and satisfy the initial conditions.

Good problem solvers go through these stages systematically, adapting their approach as needed. This structured methodical method not only improves accuracy but also fosters deeper mathematical understanding.