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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 108
Chapter 3, Problem 108

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.
(f/g)(−1)(f/g)(-1)

Verified step by step guidance
1
Identify the functions given: \(f(x) = 3x^2 - 4\) and \(g(x) = x^2 - 3x - 4\).
Understand that \((f/g)(x)\) means the function \(f(x)\) divided by \(g(x)\), so \((f/g)(x) = \frac{f(x)}{g(x)}\).
Substitute \(x = -1\) into both functions separately: calculate \(f(-1)\) and \(g(-1)\) by replacing \(x\) with \(-1\) in each expression.
Write the expression for \((f/g)(-1)\) as \(\frac{f(-1)}{g(-1)}\) using the values found in the previous step.
Simplify the fraction \(\frac{f(-1)}{g(-1)}\) by performing the arithmetic operations in the numerator and denominator, then divide to find the value of \((f/g)(-1)\).

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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 f(x), represents a rule that assigns each input x to an output. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.
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Evaluating Composed Functions

Function Division (Quotient of Functions)

The division of two functions (f/g)(x) is defined as f(x) divided by g(x), provided g(x) ≠ 0. To find (f/g)(a), evaluate both f(a) and g(a) separately, then divide the results.
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Multiplying & Dividing Functions

Polynomial Functions and Simplification

Both f(x) and g(x) are polynomial functions, which involve powers of x with coefficients. Simplifying polynomial expressions after substitution involves applying arithmetic operations and combining like terms carefully.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.

(f+g)(2k)(f+g)(2k)

Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible

(g○ƒ)(3) (g ○ ƒ)(3)

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(d) y = |ƒ(x)|

Textbook Question

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

5
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Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

(ƒ○g)(x)(ƒ ○ g)(x)

Textbook Question

Let ƒ(x)=√(x−2)ƒ(x) = √(x-2) and g(x)=x2g(x) = x^2. Find each of the following, if possible.

(g○ƒ)(x)(g ○ ƒ)(x)