Let ƒ(x)=x2+3 and g(x)=-2x+6. Find each of the following. (ƒ/g)(-1)

Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 17

Chapter 3, Problem 17

Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒ/g)(-1)

Verified step by step guidance

Understand that the notation (ƒ/g)(x) means the function ƒ(x) divided by the function g(x), so (ƒ/g)(x) = $\frac{ƒ(x)}{g(x)}$.

Substitute the given functions into the expression: (ƒ/g)(x) = $\frac{x^2 + 3}{-2x + 6}$.

Replace x with -1 in both the numerator and the denominator: numerator = (-1)^2 + 3, denominator = -2(-1) + 6.

Calculate the values inside the numerator and denominator separately: numerator = 1 + 3, denominator = 2 + 6.

Write the final expression as a fraction using these values: (ƒ/g)(-1) = $\frac{1 + 3}{2 + 6}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Notation and Evaluation

Function notation, such as ƒ(x), represents a rule that assigns each input x to an output value. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.

Function Division (Quotient of Functions)

The quotient of two functions (ƒ/g)(x) is defined as ƒ(x) divided by g(x), provided g(x) ≠ 0. To find (ƒ/g)(-1), evaluate both ƒ(-1) and g(-1) separately, then divide the results.

Domain Restrictions in Function Operations

When dividing functions, the domain excludes values where the denominator function equals zero, as division by zero is undefined. It's important to check g(x) ≠ 0 before evaluating (ƒ/g)(x) at a given point.

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Domain Restrictions of Composed Functions

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