/*! 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 22 Evaluate the given functions. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 22

Evaluate the given functions. $$T(t)=5 t+7 ; \text { find } T(2 t+a-1) \text { and } T(2 t)+a-1$$.

Short Answer

Expert verified
\( T(2t+a-1) = 10t + 5a + 2 \) and \( T(2t) + a - 1 = 10t + a + 6 \).

Step by step solution

01

Understand the Function

The given function is \( T(t) = 5t + 7 \). This function takes an input \( t \) and outputs a value based on the expression \( 5t + 7 \).
02

Substitute into the Function

First, let's find \( T(2t+a-1) \). Substitute \( 2t + a - 1 \) into the function \( T(t) = 5t + 7 \). This gives us \( T(2t + a - 1) = 5(2t + a - 1) + 7 \).
03

Simplify the Expression

Expand the expression: \( 5(2t + a - 1) = 10t + 5a - 5 \). Then add \( 7 \) to the expression: \( 10t + 5a - 5 + 7 = 10t + 5a + 2 \).
04

Evaluate \( T(2t) + a - 1 \)

First, substitute \( 2t \) into the function: \( T(2t) = 5(2t) + 7 = 10t + 7 \). Then, add \( a - 1 \) to this result: \( 10t + 7 + a - 1 = 10t + a + 6 \).
05

Conclusion: Write Final Outputs

The value of \( T(2t + a - 1) \) is \( 10t + 5a + 2 \), and the value of \( T(2t) + a - 1 \) is \( 10t + a + 6 \).

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.

Algebraic Functions
Algebraic functions form an essential part of mathematics, representing relationships between variables using algebraic expressions. In this particular case, the function \( T(t) = 5t + 7 \) is a linear function. Linear functions are expressed in the form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
  • **Slope (m):** This tells us how "steep" the function is. In \( T(t) = 5t + 7 \), the slope is 5, indicating that the output increases by 5 for every unit increase in \( t \).
  • **Intercept (b):** The value of the function when the input is zero. For \( T(t) \), when \( t = 0 \), \( T(t) = 7 \).
These characteristics make it straightforward to predict outcomes once you understand the basic form of the function. The process involves evaluating the function for different inputs and requires an understanding of how each component of the function affects the output.
Substitution Method
The substitution method is a powerful algebraic technique used to solve equations and evaluate functions. In evaluating \( T(2t + a - 1) \), we replace the input \( t \) in the original function with the expression \( 2t + a - 1 \).
  • **Identify the function:** Start with the given function, \( T(t) = 5t + 7 \).
  • **Determine the new input:** Here, the new input is \( 2t + a - 1 \).
  • **Substitute the expression:** Replace \( t \) in the original function with \( 2t + a - 1 \), giving us \( T(2t + a - 1) = 5(2t + a - 1) + 7 \).
Substitution is particularly useful as it allows you to transform functions based on varying inputs, making it versatile for solving more complex algebraic problems.
Expression Simplification
Simplification is the process of reducing an expression to its simplest form. After substituting the new input into the function, it鈥檚 crucial to simplify the resulting algebraic expression.In the example, we evaluated \( T(2t + a - 1) \) to finally compute \( 5(2t + a - 1) + 7 \). Here is how simplification works step-by-step:
  • **Distribute the 5 across each term:** Multiply each term inside the parentheses by 5: \( 5 \times 2t = 10t \), \( 5 \times a = 5a \), and \( 5 \times -1 = -5 \).
  • **Add the constants:** Combine the constants outside the parentheses: \(-5 + 7 = 2 \).
  • **Rewrite the expression:** Combine all terms to form a simplified expression: \( 10t + 5a + 2 \).
Simplification is a core skill in algebra. It transforms expressions into forms that are easier to understand and work with, providing clarity when dealing with complex algebraic expressions.

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

Solve the given problems A rectangular grazing range with an area of \(8 \mathrm{mi}^{2}\) is to be fenced. Express the length \(l\) of the field as a function of its width \(w\). What are the domain and range of \(l=f(w) ?\)

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\)

Use the following table that gives the rate R of discharge from a tank of water as a function of the height \(H\) of water in the tank. For Exercises 19 and \(20,\) plot the graph and find the values from the graph. For Exercises 21 and \(22,\) find the indicated values by linear interpolation. $$\begin{array}{l|c|c|c|c|c|c|c}\text {Height}(\mathrm{ft}) & 0 & 1.0 & 2.0 & 4.0 & 6.0 & 8.0 & 12 \\ \hline \text {Rate }\left(\mathrm{ft}^{3} / \mathrm{s}\right) & 0 & 10 & 15 & 22 & 27 & 31 & 35 \end{array}$$. (a) For \(R=34 \mathrm{ft}^{3} / \mathrm{s},\) find \(H\). (b) For \(H=6.4 \mathrm{ft}\), find \(R\).

Solve the given problems. A balloon is being blown up at a constant rate. (a) Sketch a reasonable graph of the radius of the balloon as a function of time. (b) Compare to a typical situation that can be described by \(r=\sqrt[3]{3 t}\) where \(r\) is the radius (in \(\mathrm{cm}\) ) and \(t\) is the time (in s).

Solve the given problems. The length \(L\) (in \(\mathrm{ft}\) ) of a cable hanging between equal supports \(\left.100 \mathrm{ft} \text { apart is } L=100\left(1+0.0003 s^{2}\right), \text { where } s \text { is the sag (in } \mathrm{ft}\right)\) in the middle of the cable. Because \(L=f(s),\) find \(f(15)\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision 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.