Problem 96 - College Algebra Essentials

Question

let \(f\) and \(g\) be defined by the following table: $$ \begin{array}{ccc} \hline x & f(x) & g(x) \\ \hline-2 & 6 & 0 \\ -1 & 3 & 4 \\ 0 & -1 & 1 \\ 1 & -4 & -3 \\ 2 & 0 & -6 \end{array} $$ Find \(|f(1)-f(0)|-[g(1)]^{2}+g(1) \div f(-1) \cdot g(2)\)

Step-by-Step Solution

Answer

The value of \(|f(1)-f(0)|-[g(1)]^{2}+g(1) \div f(-1) \cdot g(2)\) is 0.

Step 1: Substitution of Values

Evaluate the operations by substituting from table values: \nFind \(|f(1)-f(0)|-[g(1)]^{2}+g(1) \div f(-1) \cdot g(2)\) \nReplace \(f(1)\), \(f(0)\), \(g(1)\), \(f(-1)\), \(g(2)\) with their corresponding values from the table: \nThis results in \(|-4 - (-1)| - (-3)^{2} + (-3) \div 3 \cdot (-6)\).

Step 2: Simplify Absolute Value and Square Value

The absolute value operation makes a number positive, and square operation makes a number positive. So, this becomes \(| -3 |- 9 -1 \cdot -6\).

Step 3: Carry Out All Operations

Now we perform these operations, which will be \(3 - 9 + 6\).

Step 4: Final Computation

Finally, get the result of the last operation, which would be \(0\).

Key Concepts

Absolute Value

The absolute value of a number can be thought of as its distance from zero on the number line, no matter whether it is to the left (negative) or to the right (positive). It is always a non-negative value. In our problem, we come across the expression \(|f(1)-f(0)|\).

In mathematical terms, the absolute value is defined as:

• \(|a| = a\) if \(a\) is greater than or equal to zero,
• \(|a| = -a\) if \(a\) is less than zero.

To compute absolute values, consider an example with numbers from our exercise, \(|-4 - (-1)|\), which simplifies to \(|-3|\). Since -3 is less than zero, the absolute value is the negation of -3, meaning \(|-3| = 3\). By understanding the concept of absolute value, solving portions of the exercise that involve this operation becomes straightforward.

Substitution Method

The substitution method is an essential technique in algebra, especially when dealing with function operations. It involves replacing variables with their corresponding values. In the context of our exercise, we take values defined in a table for given functions, \(f(x)\) and \(g(x)\), and substitute them into an expression to find its value.

For instance, the expression requires us to find values for \(f(1)\), \(f(0)\), \(g(1)\), \(f(-1)\), and \(g(2)\). Referring to the table provided, we replace these with their numerical counterparts and thus bring our complex algebraic expression down to simple arithmetic. Applying the substitution method in a step-by-step manner ensures accuracy and simplifies complex problems into manageable calculations.

Function Evaluation

Function evaluation is a fundamental concept where we find the output of a function for a particular input. In algebra, to evaluate a function like \(f(x)\) for a specific value of \(x\), you simply substitute the value of \(x\) into the function and calculate.

For example, if \(f(x) = 2x + 3\) and we want to find \(f(2)\), we would substitute 2 into the function, resulting in \(f(2) = 2(2) + 3 = 7\). In our exercise, using function evaluation is vital for finding the values for \(f(1)\), \(f(0)\), \(g(1)\), and so on. Once we have the values from the function table, we need to substitute them accordingly and perform the necessary arithmetic operations, which culminates in solving the entire mathematical problem.