/*! 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 55 Find \(f(x)\) at the indicated v... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 55

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=2 x^{2}-x+3, x=1$$

Short Answer

Expert verified
The value of \( f(1) \) is 4.

Step by step solution

01

Understand the Function

The function given is \( f(x) = 2x^2 - x + 3 \). This is a quadratic function, which means it is expressed as \( ax^2 + bx + c \), where in this case \( a = 2 \), \( b = -1 \), and \( c = 3 \). Our task is to find the value of this function when \( x = 1 \).
02

Substitute the Value of x into the Function

We need to find \( f(1) \). Substitute \( x = 1 \) into the function expression: \[ f(1) = 2(1)^2 - 1 + 3 \].
03

Evaluate the Squared Term

Calculate \( 1^2 \): \[ f(1) = 2(1) - 1 + 3 \]. Since \( 1^2 = 1 \), proceed to the next step.
04

Multiply

Multiply \( 2 \) by the result from Step 3: \[ 2 \times 1 = 2 \]. Therefore, the expression becomes \[ f(1) = 2 - 1 + 3 \].
05

Perform Addition and Subtraction

Simplify the expression further:- First, \( 2 - 1 = 1 \).- Then, add \( 1 + 3 = 4 \).So, \( f(1) = 4 \).

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Key Concepts

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

Function Evaluation
In mathematics, evaluating a function means determining the output value when a specific input value is substituted into the function's equation. For example, if given the function \( f(x) = 2x^2 - x + 3 \), to evaluate this function at \( x = 1 \), you find what \( f(1) \) equals.
Function evaluation is a step-by-step process where you:
  • Identify the function and its input values.
  • Substitute the given input value (i.e., \( x = 1 \)) into the function.
  • Follow the order of operations to calculate the function's output value.
This process helps you understand how the function behaves for different input values. You can use this for quadratics, polynomials, or any type of functions.
Substitution Method
The substitution method involves replacing a variable in an equation or expression with a given value. In the quadratic function \( f(x) = 2x^2 - x + 3 \), substituting \( x = 1 \) means you are placing 1 in position of every \( x \) found in the equation.
When substituting, keep these steps in mind:
  • Write down the expression or equation clearly.
  • Carefully replace the variable \( x \) with the value provided, without altering any other part of the function.
  • Follow through with the arithmetic, maintaining the correct order of operations: parentheses, exponents, multiplication/division, and addition/subtraction (PEMDAS).
This technique simplifies the expression, allowing for straightforward calculation of the desired solution.
Polynomial Expressions
Polynomial expressions involve variables raised to different powers summed with coefficients. In this case, \( f(x) = 2x^2 - x + 3 \) is a quadratic polynomial because it features a term with \( x^2 \). Each polynomial has what we call 'terms,' for example:
  • The term \( 2x^2 \), where 2 is the coefficient and the power is 2.
  • The term \(-x \), which implies a coefficient of -1 and power of 1.
  • The constant term +3, which does not have an \( x \) variable.
Understanding polynomials is crucial because they appear frequently in algebra and calculus. They lay the groundwork for solving complex equations and modeling real-world scenarios. Recognizing and manipulating these expressions help in finding solutions efficiently.

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Most popular questions from this chapter

A linear function \(f\) has the ordered pairs listed in the table. Find the slope \(m\) of \(t F\) e table to find the \(y\) -intercept of the line, and give an equation that defines \(f\). \begin{array}{c|c} x & f(x) \\ \hline-3 & -10 \\ -2 & -6 \\ -1 & -2 \\ 0 & 2 \\ 1 & 6 \end{array}

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(4,1),(3,-5),(-2,3),(3,7)\\}$$

90\. Air Temperature When the relative humidity is \(100 \%\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. If the temperature is \(80^{\circ} \mathrm{F}\) on the ground, then \(f(x)=80-5.8 x\) calculates the air temperature \(x\) miles above the ground. Find \(f(3)\) and interpret the result. (Source: Battan, L., Weather in Your Life, W.H. Freeman.)

Using interval notation, write each set. Then graph it on a number line. $$\\{x | 8>x>3\\}$$

Rainfall By noon, 3 inches of rain had fallen during a storm. Rain continued to fall at a rate of \(\frac{1}{4}\) inch per hour. (a) Find a formula for a linear function \(f\) that models the amount of rainfall \(x\) hours past noon. (b) Find the total amount of rainfall by 2: 30 P.M.
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