Chapter 33: Problem 1
If \(f(x)=4 x^{2}-3 x+2\) find: \(f(0), f(3), f(-1)\) and \(f(3)-f(-1)\)
Short Answer
Expert verified
\(f(0) = 2\), \(f(3) = 29\), \(f(-1) = 9\), and \(f(3) - f(-1) = 20\).
Step by step solution
01
Evaluate f(0)
Substitute \(x = 0\) into the function \(f(x) = 4x^2 - 3x + 2\). This gives \(f(0) = 4(0)^2 - 3(0) + 2 = 2\). Thus, \(f(0) = 2\).
02
Evaluate f(3)
Substitute \(x = 3\) into the function. We calculate \(f(3) = 4(3)^2 - 3(3) + 2 = 4 \times 9 - 9 + 2 = 36 - 9 + 2 = 29\). So, \(f(3) = 29\).
03
Evaluate f(-1)
Replace \(x = -1\) in the function. Compute \(f(-1) = 4(-1)^2 - 3(-1) + 2 = 4 \times 1 + 3 + 2 = 4 + 3 + 2 = 9\). Therefore, \(f(-1) = 9\).
04
Calculate f(3) - f(-1)
To find \(f(3) - f(-1)\), subtract \(f(-1)\) from \(f(3)\). This gives \(f(3) - f(-1) = 29 - 9 = 20\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Evaluation
In mathematics, the evaluation of a polynomial function involves plugging specific values into the function to determine its result. When you encounter a problem that requires finding several evaluations, like evaluating a function for different values of \(x\), it's referred to as function evaluation. This process helps in understanding how a given function behaves with different inputs.
The polynomial function in this exercise is \(f(x)=4x^2-3x+2\). To evaluate a function, we simply replace the variable \(x\) with the desired value and follow through with the arithmetic operations:
The polynomial function in this exercise is \(f(x)=4x^2-3x+2\). To evaluate a function, we simply replace the variable \(x\) with the desired value and follow through with the arithmetic operations:
- For \(f(0)\), substitute \(x = 0\), leading to \(f(0) = 4(0)^2 - 3(0) + 2 = 2\).
- For \(f(3)\), substitute \(x = 3\), leading to \(f(3) = 4(3)^2 - 3(3) + 2 = 29\).
- For \(f(-1)\), substitute \(x = -1\), leading to \(f(-1) = 4(-1)^2 - 3(-1) + 2 = 9\).
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition or multiplication. In the provided exercise, the expression \(4x^2 - 3x + 2\) represents a polynomial function of \(x\). Each component of this expression has a distinct role:
- The term \(4x^2\) is called a quadratic term, since the variable \(x\) is raised to the power of 2.
- \(-3x\) is referred to as the linear term due to the presence of the variable \(x\) to the power of 1.
- Lastly, \(+2\) represents the constant term, which stays the same regardless of the value of \(x\).
Substitution Method
The substitution method, also known as plugging in values, is a fundamental technique in algebra used to evaluate expressions or solve equations. It's simply the action of replacing a variable with a numerical value or expression. This method is invaluable in simplifying and calculating the result of an expression.
Consider our polynomial \(f(x) = 4x^2 - 3x + 2\). To evaluate \(f(x)\) for specific values, we employ substitution by replacing \(x\) with numbers:
Consider our polynomial \(f(x) = 4x^2 - 3x + 2\). To evaluate \(f(x)\) for specific values, we employ substitution by replacing \(x\) with numbers:
- Place \(x = 0\) in \(f(x)\) to find \(f(0) = 2\).
- Insert \(x = 3\) to compute \(f(3) = 29\).
- Replace \(x = -1\) to get \(f(-1) = 9\).