Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 63

Chapter 3, Problem 63

In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. g (f[h (1)])

Verified step by step guidance

Step 1: Start by evaluating the innermost function, h(1). Substitute x = 1 into h(x) = x² + x + 2. This means you calculate h(1) = (1)² + (1) + 2.

Step 2: Use the result from h(1) as the input for the next function, f(x). Substitute the value of h(1) into f(x) = 2x - 5. This means you calculate f(h(1)) = 2(h(1)) - 5.

Step 3: Use the result from f(h(1)) as the input for the next function, g(x). Substitute the value of f(h(1)) into g(x) = 4x - 1. This means you calculate g(f(h(1))) = 4(f(h(1))) - 1.

Step 4: Combine all the results step by step, ensuring each function is evaluated correctly before moving to the next.

Step 5: The final result is g(f(h(1))). You now have the steps to evaluate the indicated function without finding an explicit equation for the composite function.

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Key Concepts

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

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this case, we need to evaluate g(f[h(1)]), meaning we first find h(1), then use that result as the input for f, and finally use the output of f as the input for g. Understanding how to properly nest functions is crucial for solving such problems.

Evaluating Functions

Evaluating a function means substituting a specific value into the function's equation to find the output. For example, to evaluate h(1) for the function h(x) = x² + x + 2, we replace x with 1, resulting in h(1) = 1² + 1 + 2 = 4. This step is essential for determining the values needed for further function evaluations.

Linear vs. Quadratic Functions

Linear functions, like f(x) = 2x - 5 and g(x) = 4x - 1, have a constant rate of change and graph as straight lines, while quadratic functions, such as h(x) = x² + x + 2, have a variable rate of change and graph as parabolas. Recognizing the differences between these types of functions helps in understanding their behavior and how to manipulate them during evaluations.

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Related Practice

Textbook Question

In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 8x – 4y – 12 =0

Textbook Question

Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. f(g[h (1)])

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Textbook Question

In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x + 3)

Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

Textbook Question

Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).

f(x) = 2x-3, g(x) = (x+3)/2

Textbook Question

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²

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