Question

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

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

Understanding the Functions

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

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

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

- 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: \( (1, fty) \).

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

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

- 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: \( (-1, fty) \).

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

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

- 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: \( (-fty, 1) \).

Key concepts

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.