Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 27a

Chapter 3, Problem 27a

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 a. f(6)

Verified step by step guidance

Step 1: Understand the problem. The function f(x) = 4x + 5 is given, and you are asked to evaluate it at x = 6. This means substituting 6 for x in the function.

Step 2: Substitute x = 6 into the function. Replace every occurrence of x in f(x) = 4x + 5 with 6. The expression becomes f(6) = 4(6) + 5.

Step 3: Simplify the multiplication. Multiply 4 by 6 to get 24. The expression now becomes f(6) = 24 + 5.

Step 4: Simplify the addition. Add 24 and 5 to simplify the expression further.

Step 5: The result of f(6) is the simplified value obtained in the previous step. This is the value of the function at x = 6.

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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 value for the independent variable in a function. In this case, to evaluate f(6) for the function f(x) = 4x + 5, you replace x with 6, resulting in f(6) = 4(6) + 5. This process is fundamental in understanding how functions operate and yield outputs based on given inputs.

Linear Functions

A linear function is a polynomial function of degree one, which can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The function f(x) = 4x + 5 is linear, with a slope of 4 and a y-intercept of 5. Understanding the characteristics of linear functions helps in visualizing their graphs and predicting their behavior.

Simplification

Simplification in mathematics refers to the process of reducing an expression to its simplest form. After evaluating the function, the next step is to simplify the result, if necessary. For example, in f(6) = 24 + 5, the simplification leads to f(6) = 29, which is the final output. Mastering simplification is crucial for clear and concise mathematical communication.

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Related Practice

Textbook Question

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 c. f(-x)

Textbook Question

Find the midpoint of each line segment with the given endpoints. (8, 3√5) and (−6, 7√5)

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Textbook Question

The functions in Exercises 11-28 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(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (2x +1)/(x-3)

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Textbook Question

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 b. f(x + 1)

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Textbook Question

Find the midpoint of each line segment with the given endpoints. (7√3, −6) and (3√3, −2)

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Textbook Question

Find the domain of each function. g(x) = √(x −2) /(x-5)

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