Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 63

Capitolo 3, Problema 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)])

Guida verificata passo dopo passo

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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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.

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Function Composition

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.

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Evaluating Composed Functions

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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Permutations vs. Combinations

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