Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 94a

Chapter 3, Problem 94a

Let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g(-1) and f(g(-1)).

Verified step by step guidance

Step 1: Start by evaluating g(-1). Substitute x = -1 into the function g(x) = 3x - 5. This means you will calculate g(-1) = 3(-1) - 5.

Step 2: Simplify the expression for g(-1). Multiply 3 by -1 and then subtract 5. This will give you the value of g(-1).

Step 3: Use the result from Step 2 as the input for the function f(x). Substitute g(-1) into f(x) = x² - x + 4. This means you will calculate f(g(-1)) = (g(-1))² - g(-1) + 4.

Step 4: Expand and simplify the expression for f(g(-1)). Square the value of g(-1), subtract g(-1), and then add 4.

Step 5: Combine all terms to simplify the expression for f(g(-1)). This will give you the final 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 Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, to find g(-1), we replace x in the function g(x) = 3x - 5 with -1, resulting in g(-1) = 3(-1) - 5 = -8. Understanding how to evaluate functions is crucial for solving problems that require finding specific outputs based on given inputs.

Composition of Functions

The composition of functions occurs when the output of one function becomes the input of another. In this case, after finding g(-1), we use that result as the input for the function f. The notation f(g(-1)) indicates that we first evaluate g at -1 and then substitute that output into f, which is essential for solving problems that involve multiple functions.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. In this problem, f(x) = x² - x + 4 is a quadratic function where a = 1, b = -1, and c = 4. Understanding the properties of quadratic functions, such as their parabolas and vertex, is important for analyzing their behavior and outputs.

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The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^-1(x), the inverse function. (b) Verify that your equation is correct by showing that f(f^-1(x)) = x and f^-1(f(x)) = x. f(x) = 4x - 3

Textbook Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

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