Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=2, x=0\) and \(x=-2\) $$ g(x)=x+4 $$

Step-by-Step Solution

Answer

Therefore, the values of the function at \(x=2, x=0\) and \(x=-2\) are \(g(2)=6\), \(g(0)=4\) and \(g(-2)=2\) respectively.

Step 1: Substituting \(x=2\)

To evaluate the function at \(x=2\), substitute \(x\) with 2 in the equation \(g(x) = x + 4\) to obtain \(g(2) = 2 + 4\)

Step 2: Calculation for \(x=2\)

Adding 2 and 4 yields 6, thus \(g(2) = 6\)

Step 3: Substituting \(x=0\)

To evaluate the function at \(x=0\), substitute \(x\) with 0 in the equation \(g(x) = x + 4\) to obtain \(g(0) = 0 + 4\)

Step 4: Calculation for \(x=0\)

Adding 0 and 4 yields 4, thus \(g(0) = 4\)

Step 5: Substituting \(x=-2\)

To evaluate the function at \(x=-2\), substitute \(x\) with -2 in the equation \(g(x) = x + 4\) to obtain \(g(-2) = -2 + 4\)

Step 6: Calculation for \(x=-2\)

Adding -2 and 4 yields 2, thus \(g(-2) = 2\)

Key Concepts

Substitution in Functions

Understanding substitution in functions can be very helpful and it often makes solving math problems easier. When you evaluate a function, you're essentially replacing the variable in the function with a specific number, known as substitution.

Imagine you have a function, such as "\(g(x) = x + 4\)." To find the value of \(g(x)\) when \(x = 2\), you just replace "\(x\)" with "2." This becomes \(g(2) = 2 + 4\). This process applies to any value you want to evaluate within the function. Just substitute each value into the function where the variable appears.

• Choose your specific value for the variable.
• Replace or substitute that value for the variable in the function.
• Perform the arithmetic operation indicated.

Substitution is a very straightforward step but crucial for precisely evaluating a function. Without it, it would be challenging to find the function's output for different values of \(x\). Ensure you're substituting the correct value each time to avoid mistakes.

Linear Functions

Linear functions are the simplest type of functions. A linear function looks like this: \(f(x) = mx + b\). Here, "\(m\)" represents the slope, and "\(b\)" is the y-intercept. In our specific example, the function is \(g(x) = x + 4\). While the slope \(m\) is hidden, it is actually 1.

Linear functions form a straight line when plotted on a graph. This means that their rate of change is constant, and for every step of \(x\), the change in \(y\) stays the same. Here are a few cool attributes of linear functions:

• The graph is always a straight line.
• Increasing \(x\) by 1 will increase \(y\) consistently by the value of the slope \(m\).
• The function is very predictable, which makes it simple to visualize in a graph format.

Understanding linear functions is important, since they are a stepping stone to exploring more complex functions. They also appear frequently in real-world applications such as calculating speed, budgeting, and predicting growth.

Arithmetic Operations

Arithmetic operations are the basic calculations needed to solve mathematical problems. These include addition, subtraction, multiplication, and division.

In function evaluation, arithmetic operations are used once you've substituted a value into the function—like when we evaluated \(g(x) = x + 4\). For example, when we substituted \(x = 2\) into the function, we performed the operation \(2 + 4\) to get 6.

It's important to understand how these operations interact:

• Addition: Combining values to get a sum (e.g., 2 + 4 = 6).
• Subtraction: Finding the difference between values (e.g., 4 - 2 = 2).
• Multiplication: Scaling the value by a factor (e.g., 3 * 2 = 6).
• Division: Distributing a value into equal parts (e.g., 6 / 2 = 3).

These operations form the basis of algebra and are necessary for correctly evaluating functions. Make sure you are comfortable with them before tackling more challenging math problems.