Problem 29 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=2, x=0\) and \(x=-2\) $$ f(x)=5 x+1 $$

Step-by-Step Solution

Answer

The values of the function \(f(x) = 5x + 1\) at \(x = 2, 0, -2\) are 11, 1, and -9 respectively.

Step 1: Substitution of x=2

To evaluate \(f(x)\) at \(x = 2\), substitute \(x = 2\) into \(f(x)\) to get \(f(2) = 5(2) + 1 = 11\)

Step 2: Substitution of x=0

To evaluate \(f(x)\) at \(x = 0\), substitute \(x = 0\) into \(f(x)\) to get \(f(0) = 5(0) + 1 = 1\)

Step 3: Substitution of x=-2

To evaluate \(f(x)\) at \(x = -2\), substitute \(x = -2\) into \(f(x)\) to get \(f(-2) = 5(-2) + 1 = -9\)

Key Concepts

Substitution

Substitution is a process used in mathematics where you replace a variable in an expression with a specific value or another expression. This helps you evaluate or simplify the given expression. In our given problem, we perform substitution by replacing the variable \(x\) in the function \(f(x) = 5x + 1\) with specific numeric values.

• Step 1: Substitute \(x = 2\) into the function, resulting in \(f(2) = 5 \times 2 + 1 = 11\).
• Step 2: Substitute \(x = 0\) into the function, leading to \(f(0) = 5 \times 0 + 1 = 1\).
• Step 3: Substitute \(x = -2\) into the function, yielding \(f(-2) = 5 \times (-2) + 1 = -9\).

By substituting these values, we're finding out what the output, or result, of the function is for each of these inputs. This method provides a clear way to evaluate functions at specific given points.

Linear Functions

Linear functions are a type of function that create straight lines when graphed on the coordinate plane. They follow the formula \(f(x) = mx + b\), where \(m\) and \(b\) are constants. Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
In our exercise, the function is \(f(x) = 5x + 1\).

• The slope \(m\) is 5. This indicates how steep the line is and that it rises 5 units for every 1 unit it moves to the right.
• The y-intercept \(b\) is 1, so the line crosses the y-axis at the point \((0, 1)\).

Linear functions are straightforward and predictably structured, which makes them easy to evaluate and understand. Knowing these key components allows you to quickly graph the line and understand how changes in \(x\) affect the output \(f(x)\).

Evaluation at Specific Points

Evaluating a function at specific points involves finding the output of the function, given particular input values. This helps understand how the function behaves or changes with different inputs.
In the example, we evaluated the function \(f(x) = 5x + 1\) at specific values of \(x\): \(x=2\), \(x=0\), and \(x=-2\).

• For \(x = 2\), the function's output is 11, showing how the line behaves as \(x\) increases.
• For \(x = 0\), the output is 1, confirming the y-intercept visually.
• For \(x = -2\), the function gives an output of -9, illustrating how the function decreases as \(x\) becomes negative.

By evaluating at these points, students can effectively visualize the function's graph, understand the linear relationship, and anticipate the output for other potential values of \(x\). This practice is fundamental in both mathematical learning and real-world applications, where knowing specific outcomes based on given inputs is crucial.