Problem 87 - Algebra for College Students

Question

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(0) $$

Step-by-Step Solution

Answer

-2

Step 1 - Substitute the value into the function

To evaluate the expression, substitute the value of 0 into the function f(x). The function is given as: \[ f(x) = 3x - 2 \] Hence, we substitute x with 0.

Step 2 - Calculate the expression

Now calculate the value of the expression after substitution. \[ f(0) = 3(0) - 2 \] Simplify this expression.

Step 3 - Simplify the result

Finally, simplify the expression: \[ f(0) = 0 - 2 = -2 \]

Key Concepts

substituting values

To start with the evaluation of any function, the first step is substituting the value given into the function's formula. This is like replacing a placeholder with an actual number. Here, we are given the function: \[ f(x) = 3x - 2 \] and we need to find the value of this function when x is 0. To do this, we substitute 0 for every instance of x in the formula. This step is crucial because it allows us to move forward to the actual calculation. When you substitute correctly, the function \[ f(x) = 3x - 2 \] becomes \[ f(0) = 3(0) - 2 \] which simplifies our task as it replaces the variable with the given value.

evaluating functions

Once we have substituted the value into the function, the next step is evaluating the function to get a numerical result. For our example, we now have \[ f(0) = 3(0) - 2 \]. This involves performing the operations indicated in the function with the substituted value. First, we need to handle the multiplication: 3 times 0 which equals 0. The function then simplifies to: \[ f(0) = 0 - 2 \]. This step is about performing any arithmetic operations needed to get closer to the final answer.

simplifying expressions

The final step in function evaluation involves simplifying the expression to get the simplest form of the answer. From our previous step, we have \[ f(0) = 0 - 2 \]. Simplifying this involves performing the subtraction operation, which is straightforward. We subtract 2 from 0, resulting in \[ f(0) = -2 \]. This is our final answer. Ensuring this step is correctly done is essential for arriving at an accurate solution. Simplification ties everything together and gives us the clean, manageable result we sought from our initial function evaluation process.