Problem 28
Question
Evaluate the function when \(x=2, x=0\) and \(x=-2\) $$ g(x)=-x-6 $$
Step-by-Step Solution
Verified Answer
The values of the function g(x) are \(g(2)=-8\), \(g(0)=-6\), \(g(-2)=-4\).
1Step 1: Substituting \(x=2\) into the function
Now, plug the value \(x=2\) into the function \(g(x)\), then we have \(g(2)=-(2)-6=-2-6=-8\).
2Step 2: Substituting \(x=0\) into the function
Plug the value \(x=0\) into the function \(g(x)\), then we have \(g(0)=-(0)-6=-0-6=-6\). Notice that the negative sign in front of the zero does not affect the outcome, so -0 is simply 0.
3Step 3: Substituting \(x=-2\) into the function
Plug the value \(x=-2\) into the function \(g(x)\), then we have \(g(-2)=-(-2)-6=2-6=-4\). Notice the double negative sign which simply makes the output positive.
Key Concepts
Function EvaluationSubstitute Values in FunctionsNegative Numbers Arithmetic
Function Evaluation
When you encounter a function in mathematics, think of it as a machine that takes an input and produces an output based on a specific rule. Evaluating a function is just like asking the machine what result it gives for a particular input.
For example, let's take the function from our exercise, which is described by the rule
\( g(x) = -x - 6 \). When we talk about function evaluation, we're looking to find out what happens with different values of \( x \). We want to know the output, called the function value, when \( x \) is 2, 0, and -2. Substituting these values one by one into the function's formula will provide us with the answers we seek.
For example, let's take the function from our exercise, which is described by the rule
\( g(x) = -x - 6 \). When we talk about function evaluation, we're looking to find out what happens with different values of \( x \). We want to know the output, called the function value, when \( x \) is 2, 0, and -2. Substituting these values one by one into the function's formula will provide us with the answers we seek.
Substitute Values in Functions
Substituting values into functions is a fundamental skill in algebra. It involves replacing the variable with a number and simplifying the expression to find the value of the function. Think of the variable like a placeholder for a number you choose.
Let's take the function from our exercise again, \( g(x) = -x - 6 \). When you substitute \( x = 2 \), you replace every \( x \) in the function with 2, which simplifies to \( -2 - 6 = -8 \). This process is the same for any number you plug into the function. It's important to follow the correct order of operations while substituting to ensure an accurate result.
Let's take the function from our exercise again, \( g(x) = -x - 6 \). When you substitute \( x = 2 \), you replace every \( x \) in the function with 2, which simplifies to \( -2 - 6 = -8 \). This process is the same for any number you plug into the function. It's important to follow the correct order of operations while substituting to ensure an accurate result.
Negative Numbers Arithmetic
Dealing with negative numbers can be tricky, but there are some rules that can help. When you subtract a negative, like in our step 3, it’s the same as adding a positive. This is because two negatives cancel each other out.
For example, when evaluating \( g(-2) = -(-2) - 6 \), the first term \( -(-2) \) becomes \( +2 \) because of the double negative, resulting in a final answer of \( 2 - 6 = -4 \). Remember that negative numbers represent values on the left side of zero on the number line, and understanding how they operate in arithmetic is crucial, especially when evaluating functions like \( g(x) \).
For example, when evaluating \( g(-2) = -(-2) - 6 \), the first term \( -(-2) \) becomes \( +2 \) because of the double negative, resulting in a final answer of \( 2 - 6 = -4 \). Remember that negative numbers represent values on the left side of zero on the number line, and understanding how they operate in arithmetic is crucial, especially when evaluating functions like \( g(x) \).