1.1 Q-5
Question Statement
Given the function , where and , find the values of and .
Background and Explanation
To solve this problem, we will use the fact that the function value is given for specific inputs, and . By substituting these values into the function, we create a system of equations that will help us solve for the unknowns and .
We also need to recall how to evaluate a polynomial at a specific point and how to solve a system of linear equations.
Solution
We are given the function:
Step 1: Substitute into the equation.
Using the given , substitute into the function:
Simplify:
Rearrange this equation:
Step 2: Substitute into the equation.
Now, using , substitute into the function:
Simplify:
Step 3: Solve the system of equations.
We now have the system of equations:
From equation (2), solve for :
Substitute into equation (1):
Simplify:
Solve for :
Step 4: Find the value of .
Now that we know , substitute it into :
Key Formulas or Methods Used
- Substitution to evaluate polynomials at specific points.
- Solving systems of linear equations.
Summary of Steps
- Substitute into the function to get an equation involving and .
- Substitute into the function to get another equation.
- Solve the system of equations:
- From , find .
- Substitute into the first equation and solve for .
- Substitute the value of into to find .
- The solution is and .