Problem 17 - Algebra and Trigonometry

Question

Evaluate the function at the indicated values. $$ f(x)=x^{2}-6 ; \quad f(-3), f(3), f(0), f\left(\frac{1}{2}\right), f(10) $$

Step-by-Step Solution

Answer

\(f(-3) = 3, f(3) = 3, f(0) = -6, f\left(\frac{1}{2}\right) = -\frac{23}{4}, f(10) = 94\)

Step 1: Evaluate for f(-3)

Substitute \(-3\) into the function \(f(x) = x^2 - 6\).\[f(-3) = (-3)^2 - 6\]Calculate:\[f(-3) = 9 - 6 = 3\]

Step 2: Evaluate for f(3)

Substitute \(3\) into the function \(f(x) = x^2 - 6\).\[f(3) = (3)^2 - 6\]Calculate:\[f(3) = 9 - 6 = 3\]

Step 3: Evaluate for f(0)

Substitute \(0\) into the function \(f(x) = x^2 - 6\).\[f(0) = (0)^2 - 6\]Calculate:\[f(0) = 0 - 6 = -6\]

Step 4: Evaluate for f(1/2)

Substitute \(\frac{1}{2}\) into the function \(f(x) = x^2 - 6\).\[f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - 6\]Calculate:\[f\left(\frac{1}{2}\right) = \frac{1}{4} - 6 = -\frac{23}{4}\]

Step 5: Evaluate for f(10)

Substitute \(10\) into the function \(f(x) = x^2 - 6\).\[f(10) = (10)^2 - 6\]Calculate:\[f(10) = 100 - 6 = 94\]

Key Concepts

Quadratic Function

A quadratic function is a type of function where the highest power of the variable is squared. In mathematical terms, it is expressed as:

• The general form: \(f(x) = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants.
• In our example, the quadratic function is \(f(x) = x^2 - 6\). Here, \(a = 1\), \(b = 0\), and \(c = -6\).

Quadratic functions are commonly represented by parabolas when graphed on a coordinate plane. The shape of the parabola can vary based on the values of \(a\), \(b\), and \(c\). For instance, if \(a > 0\), the parabola opens upwards, whereas if \(a < 0\), it opens downwards.

Understanding the components of a quadratic function is crucial, as they determine the position, direction, and shape of the graph.

Substitution

Substitution is a method used to replace a variable with a specific number. It's a critical step in evaluating functions, particularly quadratic functions. Here's how to perform substitution:

• Identify the variable in the function equation. In our case, the variable is \(x\).
• Replace the variable with the given value. For instance, substituting \(-3\) into \(f(x) = x^2 - 6\) involves replacing \(x\) with \(-3\).

This process of substitution helps to simplify the function into a form that can be easily calculated. It's an essential skill when working with functions, as it enables you to find specific outputs based on your inputs.

Once you've substituted the value, you can then follow through with any calculations to determine the function's value for that particular input.

Evaluating Function Values

Evaluating function values means calculating the output of a function for given inputs, after substitution. With quadratic functions like \(f(x) = x^2 - 6\), evaluating involves:

• Perform the arithmetic operations after substitution, such as squaring the number and then carrying out additional calculations defined by the function.
• In Step 1 of the solution, for \(f(-3)\), substitute \(-3\) to find: \((-3)^2 - 6 = 9 - 6 = 3\).
• Repeat the substitution and calculation process for each input value: \(x = 3, 0, \frac{1}{2}, 10\).

Through this process, you determine the value of the function for each specified input, effectively evaluating the function. This precise step is vital for understanding how different inputs affect the function output and how those outputs can be represented graphically.

Step-by-Step Solution

Solving a function step-by-step is a methodical approach to ensure you understand each part of the problem. It involves tackling each element of the process separately and in order. Here's why this approach is helpful:

• Clarity: Breaking down each step allows for clear comprehension of what is being done at every stage.
• Accuracy: Focusing on one step at a time reduces the chance of errors.
• Understanding: Seeing the steps laid out plainly helps reinforce learning, especially when dealing with recurring processes like evaluating quadratic functions.

For our example function \(f(x) = x^2 - 6\), the solution involves substituting each input individually and computing the output systematically. Each step ensures that the substitution is accurate and the calculations correctly display the value of the function. This technique not only aids in solving the current problem but builds a strong foundation for handling more complex problems in the future.