Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 93

Chapter 3, Problem 93

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

Verified step by step guidance

First, identify the given functions: \(f(x) = x^{2} - x + 4\) and \(g(x) = 3x - 5\).

Calculate \(g(1)\) by substituting \(x = 1\) into the function \(g(x)\): \(g(1) = 3(1) - 5\).

Simplify the expression for \(g(1)\) to find its value.

Next, find \(f(g(1))\) by substituting the value of \(g(1)\) into the function \(f(x)\): \(f(g(1)) = (g(1))^{2} - g(1) + 4\).

Simplify the expression for \(f(g(1))\) by squaring \(g(1)\), subtracting \(g(1)\), and then adding 4 to get the final expression.

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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 the function's formula to find the corresponding output. For example, to find g(1), replace x with 1 in g(x) = 3x - 5 and simplify.

Composite Functions

A composite function like f(g(1)) means applying one function to the result of another. First, evaluate the inner function g(1), then use that output as the input for the outer function f.

Polynomial and Linear Functions

Understanding the types of functions involved helps in evaluation. Here, f(x) is a quadratic polynomial (degree 2), and g(x) is a linear function (degree 1), which affects how you substitute and simplify expressions.

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Related Practice

Textbook Question

Use the graphs of f and g to evaluate each composite function.

(fog) (-1)

Textbook Question

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2

Textbook Question

In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

Textbook Question

Use the graphs of f and g to evaluate each composite function.

(go f) (0)

Textbook Question

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

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|x+3|