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Chapter 9: Problem 29

Given \(f(x)=2 x^{2}-5 x+3,\) find each of the following. $$ f(0) $$

Short Answer

Expert verified
f(0) = 3

Step by step solution

01

- Substitute the Value of x

To find the value of the function when given a specific input, substitute the given value of x into the function. Here, we need to find \(f(0)\), so substitute 0 for x in the function \(f(x) = 2x^2 - 5x + 3\).
02

- Perform the Substitution

Substitute 0 for x in \(f(x) = 2x^2 - 5x + 3\): \[ f(0) = 2(0)^2 - 5(0) + 3 \]
03

- Simplify the Expression

Calculate the expression: \[ f(0) = 2(0)^2 - 5(0) + 3 \] This simplifies to \[ f(0) = 0 - 0 + 3 \] Therefore, \[ f(0) = 3 \].

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Key Concepts

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

Substitution
Substitution is a critical method in algebra and calculus for evaluating functions. In essence, it means replacing a variable with a given number or expression. This step allows us to transform a general function into a more specific value. Imagine you have a function, which is just a fancy way to describe a relationship between variables. For instance, with the function given, \(f(x) = 2x^2 - 5x + 3\), if we are asked to find \(f(0)\), the first thing to do is substitute \(0\) in place of every \(x\) in the equation. This is how we begin to solve for specific values that the function might take on.
Polynomial Functions
Polynomial functions are a type of mathematical expression that involve sums of powers of variables. They are general expressions of the form \(a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0\), where each \(a_i\) is a constant, and \(n\) is a non-negative integer. In our example, \(2x^2 - 5x + 3\), we have a second-degree polynomial because the highest power of \(x\) is \(x^2\). Understanding polynomial functions is key because they form the basis for much of algebra and calculus. Polynomials can be easily manipulated using basic arithmetic operations and they show up in numerous mathematical and real-world contexts.
Simplification
Simplification refers to the process of reducing a mathematical expression to its most compact or straightforward form. This is often the last step after performing substitutions and operations. In our solved exercise, once we substitute \(0\) for \(x\), we get \(f(0) = 2(0)^2 - 5(0) + 3\). We then proceed to simplify it step-by-step:
  • Calculate \(2(0)^2\) which gives \(0\).
  • Calculate \(-5(0)\) which also gives \(0\).
  • Add these results together, \(0 + 0 + 3\) which leaves us with \(3\).
Essentially, simplification makes the answer easier to understand and ensures that it's in the simplest form. It's a critical skill in mathematics, reducing errors and making it easier to work with complex expressions.

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