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Chapter 2: Problem 65

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$f(a)$$

Short Answer

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

Step by step solution

01

Understand the function

Given the function \( f(x) = 3x^2 - x \), identify what needs to be calculated, which is \( f(a) \).
02

Replace the variable

To find \( f(a) \), replace every occurrence of \( x \) in the function \( f(x) \) with \( a \). This gives us the new function \( f(a) = 3a^2 - a \).
03

Simplify if necessary

In this case, the result is already simplified. Therefore, \( f(a) = 3a^2 - a \) is the final answer.

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving variables and coefficients, combined using addition, subtraction, and multiplication. The power of the variable (often denoted as x) in polynomial functions is always a non-negative integer.

In the given problem, we have two polynomial functions: \(f(x) = 3x^2 - x\) and \(g(x) = 4x - 2\). Here are a few key points about polynomial functions:
  • Each term in a polynomial has a coefficient (a constant number) and a variable raised to a power.
  • The degree of a polynomial is the highest power of the variable in the expression.

For example, in \(f(x) = 3x^2 - x\), the degree is 2 (because of the term \(3x^2\)), and the coefficients are 3 and -1.
Substitution Method
The substitution method involves replacing a variable in an expression with another value. This is a critical skill in algebra, particularly for evaluating functions.

For the function \(f(x) = 3x^2 - x\), to find \(f(a)\), you substitute every occurrence of \(x\) with \(a\). Let鈥檚 break it down:
  • Start with the given expression: \(f(x) = 3x^2 - x\).
  • Replace \(x\) with \(a\) to get \(f(a) = 3a^2 - a\).
This method allows you to evaluate the function at any given point by substituting different values for the variable.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. This often involves combining like terms and ensuring there are no unnecessary operations.

In the provided exercise, after substituting \(x\) with \(a\) in \(f(x) = 3x^2 - x\), we get \(f(a) = 3a^2 - a\). Here, the expression is already in its simplest form.

Here are some tips for simplifying expressions:
  • Combine like terms (terms with the same variable and exponent).
  • Use basic arithmetic operations to reduce the expression.
  • Factorize when possible to make the expression more manageable.
By simplifying the expression, you make the function easier to work with for further calculations or for understanding the behavior of the function.

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Most popular questions from this chapter

Solve each problem. Across-the-Board Raise Each teacher at C. F. Gauss Elementary School is given an across-the-board raise of 2000 dollar. Write a function that transforms each old salary \(x\) into a new salary \(N(x)\)

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{4}-x^{3}$$

Use transformations to graph each function and state the domain and range. $$y=3|x-2|$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\sqrt{x}$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{3}-5 x+1$$
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