/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 (a) \(f(3)=-5\). (b) \(f(x) \e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

(a) \(f(3)=-5\). (b) \(f(x) \equiv x^{2}-6 x+4\)

Short Answer

Expert verified
The function \(f(x)\) is \(x^2 - 6x + 4\), and \(f(3) = -5\).

Step by step solution

01

- Understand the Given Function

The given function is a quadratic function: \[f(x) = x^2 - 6x + 4\]. This is the function we will use for our calculations.
02

- Substitute x in the Function for Part (a)

To find \(f(3)\), substitute 3 into the function.Substitute \(x = 3\) into \(f(x) = x^2 - 6x + 4\): \[f(3) = (3)^2 - 6(3) + 4\]
03

- Simplify the Expression

Now, simplify the expression:\[f(3) = 9 - 18 + 4\]Combine the terms:\[f(3) = -9 + 4 = -5\]
04

- Verify the Result

According to the problem statement, \(f(3) = -5\). Our calculation also gives \(f(3) = -5\), which confirms that our function is correct.

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 91影视!

Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial function that can be represented as: \[f(x) = ax^2 + bx + c\]. It is called quadratic because the highest power of the variable \(x\) is 2 (i.e., \(x^2\)). This specific form makes quadratic functions distinct and easy to recognize.

Some key characteristics of quadratic functions include:
  • They form a parabola when graphed on a coordinate plane.
  • The coefficient \(a\) determines the direction of the parabola (upward if \(a > 0\) and downward if \(a < 0\)).
  • The vertex of the parabola is its highest or lowest point depending on the value of \(a\).
Understanding the basic structure and characteristics of quadratic functions is essential in solving related problems.
Function Evaluation
Function evaluation is the process of finding the output value of a function for a specific input value. Given a function, you substitute the input value into the function and simplify to find the result.

Consider the quadratic function given in the problem: \[f(x) = x^2 - 6x + 4\]. To evaluate this function at a particular value of \(x\), such as \(x = 3\), you carry out the following steps:
  • Substitute the value of \(x = 3\) into the function.
  • Simplify the resulting expression by performing any arithmetic operations like addition, subtraction, multiplication, or division.

In our example, we perform these steps to find \(f(3)\). This process ensures that you get the correct output for the given input, helping you understand how the function behaves for different values.
Substitution Method
The substitution method is a powerful tool in function evaluation. It involves replacing the variable in the function with a given value and then simplifying the expression to find the answer. Here鈥檚 how it works:

Given \(f(x) = x^2 - 6x + 4\) and asked to find \(f(3)\):
  • Substitute \(x = 3\) into the function: \(f(3) = (3)^2 - 6(3) + 4\).
  • Perform the simplification:
    Calculate \((3)^2\), which is 9.
    Then, calculate \(-6 \times 3\), which is -18.
    Finally, add 4: \[f(3) = 9 - 18 + 4\].
  • Combine the terms to get: \[f(3) = -5\].

This step-by-step process helps ensure accuracy and reinforces the understanding of how substitution works within functions. It鈥檚 a foundational skill for evaluating not just quadratic functions but any type of function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Express the function \(\frac{7 x+4}{(x-3)(x+2)^{2}}\) as the sum of three partial fractions with numerators independent of \(x\). \((\mathrm{JMB}) \mathrm{p}\)

\(f(x) \equiv \frac{2 x}{(x+2)(x-2)}\). (a) \(f(x)\) is an improper fraction. (b) \(f(0)=2\) (c) \(f(x) \equiv \frac{1}{x-2}+\frac{1}{x+2}\)

If \(x^{2}+4 x+p \equiv(x+q)^{2}+1\), the values of \(p\) and \(q\) are: (a) \(p=5, q=2\) (b) \(p=1, q=2\) (c) \(p=2, q=5\) (d) \(p=-1, q=5\) (e) \(p=0, q=-1\).

Write down the value of \(f(2)\) (a) \(f(X)\) is a polynomial of degree \(1 .\) (b) \(f(0)=1\) (c) \(f(1)=2\)

Given that the roots of the equation \(a x^{2}+b x+c=0 \quad\) are \(\beta\) and \(n \beta\), show that \((n+1)^{2} a c=n b^{2}\) (U of \(\mathrm{L}\) )
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.