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Chapter 5: Problem 83

If \(f(x)=x^{2}-3 x,\) find the following. See Example 12 $$ f(a) $$

Short Answer

Expert verified
\(f(a) = a^2 - 3a\)

Step by step solution

01

Understand the Function

We are given the function \(f(x) = x^2 - 3x\). We are asked to find \(f(a)\), which means we need to substitute \(a\) in place of \(x\) in the function.
02

Substitute 'a' into the Function

Replace \(x\) with \(a\) in the function \(f(x) = x^2 - 3x\). This gives us \(f(a) = a^2 - 3a\).
03

Simplify the Expression

The expression \(f(a) = a^2 - 3a\) cannot be simplified further, as it is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a fundamental concept used in mathematics to replace a variable with a specific value or another expression. In the context of function evaluation, substitution involves plugging a given value into a function in place of its variable to find the output.
In this exercise, our function is defined as \( f(x) = x^2 - 3x \). The problem requires us to determine \( f(a) \), which means we substitute \( a \) for \( x \).
  • Identify the variable in the given function.
  • Substitute the specified value for the variable throughout the entire function.
  • Ensure that all operations align correctly after substitution.
Substitution transforms \( f(x) = x^2 - 3x \) into \( f(a) = a^2 - 3a \), allowing us to evaluate the function for the specified input.
Simplification
After substituting a specific value into an expression, simplification is the process used to reduce it to its most straightforward form without changing its value. This makes expressions easier to understand and work with.

Simplification often involves:
  • Combining like terms, which are terms with the same variable raised to the same power.
  • Reducing arithmetic expressions to their simplest numeric value.
  • Removing or cancelling out any unnecessary factors or terms.
In our example, once we've substituted \( a \) for \( x \), we get \( f(a) = a^2 - 3a \). Here, the expression is already simplified because there are no like terms to combine and no additional operations to perform.
Therefore, understanding when an expression is simplified is just as important as performing the simplification.
Quadratic Function
A quadratic function is a polynomial function of degree two. The general form is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

Quadratic functions have several unique properties:
  • The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).
  • The vertex of the parabola represents the minimum or maximum point of the function.
  • Quadratics are symmetric around their vertical axis through the vertex.
In this exercise, the function \( f(x) = x^2 - 3x \) is a specific case of a quadratic function, where \( a = 1 \), \( b = -3 \), and \( c = 0 \). This parabolic shape means the function encapsulates a series of squares and linear terms, resulting in a savvy structure for modeling various real-world scenarios. Understanding this interplay between terms enables better grasp of algebraic problem-solving in quadratic contexts.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ 2 x^{2 n}+11 x^{n}+5 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}-x-12 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}+7 x^{n}-18 $$

Use substitution to factor each polynomial completely. See Examples 11 and 12. $$ (3 x-1)^{2}+5(3 x-1)+6 $$

Factor each trinomial. See Examples 5 through 10. $$ 28 y^{2}+22 y+4 $$
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