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Chapter 3: Problem 115

If \(f(x)=2 x-9\) and \(g(x)=\sqrt{x},\) find \((f+g)(9)\)

Short Answer

Expert verified
(f+g)(9) = 12

Step by step solution

01

Understand the Function Definitions

Given functions are: \(f(x) = 2x - 9\) \(g(x) = \sqrt{x}\) We need to find \((f+g)(9)\).
02

Evaluate Each Function Separately at x = 9

First, evaluate \(f(9)\): \(f(9) = 2(9) - 9 = 18 - 9 = 9\). Next, evaluate \(g(9)\): \(g(9) = \sqrt{9} = 3\).
03

Add the Results of the Functions

Now, combine the evaluated results: \((f + g)(9) = f(9) + g(9) = 9 + 3 = 12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Functions
Evaluating functions is about finding the value of a function for a given input. You start by substituting the given input value into the function.
For example, if you have a function \(f(x) = 2x - 9\), and you are asked to find \(f(9)\), you substitute 9 wherever you see \(x\) in the function. So, \(f(9) = 2(9) - 9 = 18 - 9 = 9\).
Similarly, for \(g(x) = \sqrt{x}\) and you are asked to find \(g(9)\), you substitute 9 into the function, and you get \(g(9) = \sqrt{9} = 3\).
This way, we can evaluate any function as long as we know the input value.
Composite Functions
Composite functions involve combining two or more functions. This is often done by substituting one function into another.
For instance, if we have two functions \(f(x)\) and \(g(x)\), the composite function \((f \circ g)(x)\) is created by evaluating \(g(x)\) first, and then using that result as the input for \(f(x)\).
If \(f(x) = 2x - 9\) and \(g(x) = \sqrt{x}\), then \((f \circ g)(x) = f(\sqrt{x})\).You would first find \(\sqrt{x}\), and then substitute that into the \(f(x)\) function.
Composing functions can help simplify complex calculations and can be used for more advanced problem-solving in mathematics.
Linear Functions
Linear functions are functions of the form \(f(x) = mx + b\) where m and b are constants.
These functions produce a straight line when plotted on a graph. The coefficient \(m\) represents the slope of the line, and \(b\) is the y-intercept or the point where the line crosses the y-axis.
For example, if \(f(x) = 2x - 9\), it is a linear function where the slope \(m = 2\) and the y-intercept \(b = -9\).
Linear functions are easy to evaluate, and they play a crucial role in fields like economics, physics, and engineering because they model relationships with a constant rate of change.
Square Root Function
A square root function is a function that involves the square root of the input variable, generally represented as \(g(x) = \sqrt{x}\).
For non-negative values of x, the square root function produces a non-negative output. It graphically represents a curve increasing at a decreasing rate.
For example, evaluating \(g(9)\) where \(g(x) = \sqrt{x}\), you get \(g(9) = \sqrt{9} = 3\).
The square root function is used in various applications like physics and engineering to model phenomena involving rates of change and scaling.

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Most popular questions from this chapter

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