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Chapter 10: Problem 53

If \(f(t)=4-\sqrt{t}-3\) and \(g(t)=(t+5)^{1 / 2},\) find any \(t\) for which \(f(t)=g(t)\).

Short Answer

Expert verified
There is no solution because the square root cannot be negative.

Step by step solution

01

Set the Functions Equal

To find the value of t for which the functions are equal, set the expressions for both functions equal to each other: \[ 4 - \sqrt{t} - 3 = (t + 5)^{1/2} \]
02

Simplify the Equation

Combine like terms to simplify the equation. \[ 1 - \sqrt{t} = (t + 5)^{1/2} \]
03

Isolate the Square Roots

Square both sides to eliminate the square root. \[ (1 - \sqrt{t})^2 = (t + 5) \] This expands to: \[ 1 - 2\sqrt{t} + t = t + 5 \]
04

Simplify and Solve for t

Simplify the resulting equation: \[ 1 - 2\sqrt{t} + t = t + 5 \] Combine like terms: \[ 1 - 2\sqrt{t} = 5 \] Solve for \sqrt{t}: \[ -2\sqrt{t} = 4 \] \[ \sqrt{t} = -2 \] Since a square root cannot be negative, there is no solution to this equation.

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Key Concepts

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

Functions
In mathematics, a **function** is a relation between a set of inputs and a set of permissible outputs. Each input is related to exactly one output. Functions can be thought of as machines that take an input, process it in some way, and then produce an output. For the given problem, we have two functions:
  • \f(t) = 4 - \sqrt{t} - 3
  • g(t) = (t + 5)^{1/2}
To solve for the variable \(t\) when these two functions are equal, we need to find the value of \(t \) where the outputs of both functions are the same.
Square Roots
A **square root** of a number \( x \) is a value that, when multiplied by itself, gives \( x \). It is denoted as \( \sqrt{x} \). For example, \( \sqrt{9} = 3 \) because \( 3 * 3 = 9 \). Square roots only produce non-negative results.
In the given problem, we encounter square roots in both functions:
  • \(\sqrt{t} \) in \( f(t) \)
  • (t + 5)^{1/2} in \( g(t) \)
To eliminate the square root during equation solving, we often square both sides of the equation to transform the square root into a more solvable polynomial form.
Equation Solving
To solve an **equation**, we need to find the value of the unknown variable that makes the equation true. This often involves isolating the unknown or transforming the equation to simplify it. For the exercise, we started with the equation:
  • 4 - \( \sqrt{t} \) - 3 = (t + 5)^{1/2}
Simplified to:
  • 1 - \( \sqrt{t} \) = (t + 5)^{1/2}
The next steps were squaring both sides to get rid of the square roots and then solving for \( t \). This process of isolating the variable and simplifying through algebraic manipulations is essential for solving equations.
No Solution
In some cases, equations have **no solution**. This means that there is no possible value of the variable that can satisfy the equation.
In our exercise, after simplifying and solving, here's what we found:
  • \( \sqrt{t} = -2 \)
Since the square root of a number cannot be negative, there cannot be any value of \( t \) that makes this equation true. Therefore, the equation has no solution. It is essential to recognize and understand when and why certain equations do not have solutions to correctly interpret and solve mathematical problems.

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Most popular questions from this chapter

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