/*! 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 38 Evaluate the given quantities ap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 38

Evaluate the given quantities applying the following four functions: \(f(x)=2 x-3\) \(F(t)=4-t^{2}\) \(g(t)=5+t\) \(G(x)=x^{2}+2 x-7\) $$F(-1)$$

Short Answer

Expert verified
The value of \(F(-1)\) is 3.

Step by step solution

01

Identify the Function

The given exercise requires us to evaluate the function \(F(t)\) at the specific point \(t = -1\). The function is given as \(F(t) = 4 - t^2\). We need to substitute \(-1\) in place of \(t\) in this function.
02

Substitute the Value

Replace the variable \(t\) with \(-1\) in the function \(F(t) = 4 - t^2\). This gives us the equation \(F(-1) = 4 - (-1)^2\).
03

Calculate the Square

Calculate the square of \(-1\). Since \((-1)^2 = 1\), the equation now becomes \(F(-1) = 4 - 1\).
04

Perform the Subtraction

Subtract the result of the square from 4. Thus, \(F(-1) = 4 - 1 = 3\).

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.

Substitution in Functions
One of the most fundamental operations when working with functions is substitution. Function evaluation often involves plugging a specific value into the function to see what output it produces. For example, if you have the function \(F(t) = 4 - t^2\) and you need to find \(F(-1)\), you'll perform substitution to replace the variable \(t\) with the value \
  • First, identify the function and the variable to substitute.
  • Next, substitute the given value. In our example, \(t\) is substituted with \(-1\), forming the equation \(F(-1) = 4 - (-1)^2\).
  • Check your substituted expression to ensure it's correctly formed before proceeding with further calculations.
By practicing substitution, you can efficiently evaluate any function at given points and better understand how it behaves with different inputs.
Squaring Numbers
Squaring involves multiplying a number by itself, and it is a common operation in various math problems, especially in function evaluations. In our example, after substituting \(t\) with \(-1\) in the function \(F(t) = 4 - t^2\), you need to square the substituted number.
  • The operation \((-1)^2\) requires multiplying \(-1\) by itself, resulting in \(1\). Remember, a negative number squared always results in a positive because the negatives cancel out.
  • This squared value replaces the squared variable in your original function, leading us to \(F(-1) = 4 - 1\).
Understanding how to accurately square numbers helps in preventing errors, especially with negative values, ensuring you progress smoothly through mathematical processes.
Function Operations
Function operations involve manipulating the expressions within a function. In the prior example, after squaring the substituted value in the function \(F(t) = 4 - t^2\), you would perform the operation to find \(F(-1)\).
  • The function \(F(t) = 4 - t^2\) simplifies to \(F(-1) = 4 - 1\) after substitution and squaring.
  • From there, perform the arithmetic operation, which is subtraction in this case. Subtract \(1\) from \(4\) to get \(3\).
  • This operation gives the final outcome of the function evaluation鈥擻(F(-1) = 3\).
Mastering function operations includes understanding various arithmetic processes required to finalize evaluations and helps in refining your problem-solving techniques.

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

A couple are about to put their house up for sale. They bought the house for \(\$ 172,000\) a few years ago; if they list it with a realtor, they will pay a \(6 \%\) commission. Write a function that represents the amount of money they will make on their home as a function of the asking price \(p\)

Explain the mistake that is made. Given the function \(f(x)=\sqrt{x-2},\) find the inverse function \(f^{-1}(x),\) and state the domain restrictions on \(f^{-1}(x)\) Solution: Step 1: Let \(y=f(x)\). \(\quad y=\sqrt{x-2}\) Step 2: Interchange \(x\) and \(y . \quad x=\sqrt{y-2}\) Step 3: Solve for \(y\) \(y=x^{2}+2\) Step \(4: \operatorname{Let} f^{-1}(x)=y . \quad f^{-1}(x)=x^{2}+2\) Step 5: Domain restrictions \(f(x)=\sqrt{x-2}\) has the domain restriction that \(x \geq 2\) The inverse of \(f(x)=\sqrt{x-2}\) is \(f^{-1}(x)=x^{2}+2\) The domain of \(f^{-1}(x)\) is \(x \geq 2\) This is incorrect. What mistake was made?

A \(20 \mathrm{ft} \times 10\) ft rectangular pool has been built. If 50 cubic feet of water is pumped into the pool per hour, write the water-level height (feet) as a function of time (hours).

Graph the functions \(f\) and \(g\) and the line \(y=x\) in the same screen. Do the two functions appear to be inverses of each other? $$f(x)=\sqrt[3]{x+3}-2 ; \quad g(x)=x^{3}+6 x^{2}+12 x+6$$

Using a graphing utility, plot \(y_{1}=\sqrt{x^{2}-3 x-4}\) \(y_{2}=\frac{1}{x^{2}-14},\) and \(y_{3}=\frac{1}{y_{1}^{2}-14} .\) If \(y_{1}\) represents a function \(f\) and \(y_{2}\) represents a function \(g,\) then \(y_{3}\) represents the composite function \(g \circ f .\) The graph of \(y_{3}\) is only defined for the domain of \(g \circ f .\) State the domain of \(g \circ f\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics 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.