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Chapter 10: Problem 5

Let \(f(x)=2^{x}\) and \(g(x)=1 .\) Graph each of the following, stating its domain and range. a) \(y=(f+g)(x)\) b) \(y=(f-g)(x)\) c) \(y=(g-f)(x)\)

Short Answer

Expert verified
a) Domain: \((-fty, fty)\), Range: \((1, fty)\); b) Domain: \((-fty, fty)\), Range: \((-1, fty)\); c) Domain: \((-fty, fty)\), Range: \((-fty, 1)\)

Step by step solution

01

Understanding the Functions

Given two functions: \(f(x) = 2^x\) and \(g(x) = 1\).
02

- Finding \((f+g)(x)\)

Compute \( (f+g)(x) \): \[(f+g)(x) = f(x) + g(x) = 2^x + 1\]Graph this function.
03

- Domain and Range of \((f+g)(x)\)

Domain: All real numbers \((-fty, fty)\). Range: For any real \( x \), \( 2^x \) is always positive and greater than zero, hence \( 2^x + 1 \) will be greater than 1. Therefore, Range: \( (1, fty) \).
04

- Finding \((f-g)(x)\)

Compute \((f-g)(x)\): \[(f-g)(x) = f(x) - g(x) = 2^x - 1\]Graph this function.
05

- Domain and Range of \((f-g)(x)\)

Domain: All real numbers \((-fty, fty)\). Range: Since \( 2^x \) is always positive and greater than zero, the smallest value \( 2^x - 1 \) can take when \(x \rightarrow -fty\) is -1.Therefore, Range: \( (-1, fty) \).
06

- Finding \((g-f)(x)\)

Compute \((g-f)(x)\): \[(g-f)(x) = g(x) - f(x) = 1 - 2^x\]Graph this function.
07

- Domain and Range of \((g-f)(x)\)

Domain: All real numbers \((-fty, fty)\). Range: Since \( 2^x \) grows exponentially, the smallest value of \(1 - 2^x\) occurs when \(x \rightarrow fty\), approaching \(-fty\). The maximum value is 1 when \(x = 0\).Therefore, Range: \( (-fty, 1) \).

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Key Concepts

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

function operations
Function operations involve combining functions using arithmetic operations such as addition, subtraction, multiplication, and division. When we have functions like the given \(f(x) = 2^x\) and \(g(x) = 1\), we can create new functions by adding or subtracting them.

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Most popular questions from this chapter

Let \(f_{1}(x)=x, f_{2}(x)=\frac{1}{x}, f_{3}(x)=1-x\), \(f_{4}(x)=\frac{x}{x-1}, f_{5}(x)=\frac{1}{1-x},\) and \(f_{6}(x)=\frac{x-1}{x}\). a) Determine the following. i) \(f_{2}\left(f_{3}(x)\right)\) ii) \(\left(f_{3} \circ f_{5}\right)(x)\) iii) \(f_{1}\left(f_{2}(x)\right)\) iv) \(f_{2}\left(f_{1}(x)\right)\) b) \(f_{6}^{-1}(x)\) is the same as which function listed in part a)?

Let \(j(x)=x^{2}\) and \(k(x)=x^{3} .\) Does \(k(j(x))=j(k(x))\) for all values of \(x ?\) Explain.

Let \(f(x)=\sqrt{36-x^{2}}\) and \(g(x)=\sin x\) a) Graph \(f(x), g(x),\) and \(y=(f \cdot g)(x)\) on the same set of axes. b) State the domain and range of the combined function. c) Graph \(y=\left(\frac{f}{g}\right)(x)\) and state its domain and range. d) Explain how the domain and range for \(y=\left(\frac{g}{f}\right)(x)\) differs from the domain and range in part c).

Consider \(f(x)=x^{2}-9\) and \(g(x)=\frac{1}{x}\) a) State the domain and range of each function. b) Determine \(h(x)=f(x)+g(x)\). c) How do the domain and range of each function compare to the domain and range of \(h(x) ?\).

During a race in the Sportsman category of drag racing, it is common for cars with different performance potentials to race against each other while using a handicap system. Suppose the distance, \(d_{1},\) in metres, that the faster car travels is given by \(d_{1}(t)=10 t^{2},\) where \(t\) is the time, in seconds, after the driver starts. The distance, \(d_{2},\) in metres, that the slower car travels is given by \(d_{2}(t)=5(t+2)^{2},\) where \(t\) is the time, in seconds, after the driver of the faster car starts. Write a function, \(h(t)\) that gives the relative distance between the cars over time.
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