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Chapter 9: Problem 16

For pair of functions, find (a) \((f \circ g)(1)\) (b) \((g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)\). \(f(x)=10-x ; g(x)=\sqrt{x}\)

Short Answer

Expert verified
For (a), \((f \circ g)(1) = 9\); for (b), \((g \circ f)(1) = 3\); for (c), \((f \circ g)(x) = 10 - \sqrt{x}\); for (d), \((g \circ f)(x) = \sqrt{10 - x}\)

Step by step solution

01

Understand the Functions

The given functions are:\( f(x) = 10 - x \)\( g(x) = \sqrt{x} \).
02

Find \((f \circ g)(1)\)

\((f \circ g)(x)\) means \(f(g(x))\). To find \((f \circ g)(1)\), calculate \(g(1)\) first:\( g(1) = \sqrt{1} = 1 \).Now substitute this into \(f\):\( f(g(1)) = f(1) = 10 - 1 = 9 \).
03

Find \((g \circ f)(1)\)

\((g \circ f)(x)\) means \(g(f(x))\). To find \((g \circ f)(1)\), calculate \(f(1)\) first:\( f(1) = 10 - 1 = 9 \).Now substitute this into \(g\):\( g(f(1)) = g(9) = \sqrt{9} = 3 \).
04

Find \((f \circ g)(x)\)

\((f \circ g)(x)\) means \(f(g(x))\):\( g(x) = \sqrt{x} \).Substitute \(g(x)\) into \(f\):\( f(\sqrt{x}) = 10 - \sqrt{x} \).Therefore, \((f \circ g)(x) = 10 - \sqrt{x} \).
05

Find \((g \circ f)(x)\)

\((g \circ f)(x)\) means \(g(f(x))\):\( f(x) = 10 - x \).Substitute \(f(x)\) into \(g\):\( g(10 - x) = \sqrt{10 - x} \).Therefore, \((g \circ f)(x) = \sqrt{10 - x} \).

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Key Concepts

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

composite functions
Composite functions combine two functions into one. It involves substituting one function into another. For example, with functions 饾憮(饾懃)=10鈭掟潙� and 饾憯(饾懃)=鈭氿潙�, the composite functions are (饾憮鈭橉潙�)(饾懃) and (饾憯鈭橉潙�)(饾懃).
- (饾憮鈭橉潙�)(饾懃) means we first apply 饾憯(饾懃), then apply 饾憮 to the result.
- Conversely, (饾憯鈭橉潙�)(饾懃) means we first apply 饾憮(饾懃), then apply 饾憯.
This process helps us understand how different functions interact and transform inputs.
function notation
Function notation is a way to represent functions mathematically. It is compact and clear. For the functions given:
- 饾憮(饾懃) = 10 - 饾懃: Here, 饾憮 denotes the function and 饾懃 represents the input variable.
- 饾憯(饾懃) = 鈭氿潙�: In this case, 饾憯 is the function name and 鈭氿潙� shows the operation done on 饾懃.
When we read function notation, 饾憮(饾懃) means '饾憮' of 饾懃, where we substitute 饾懃 in the function's formula to find the output. This notation is essential for evaluating and combining functions.
algebraic operations
Algebraic operations are used to combine and manipulate functions. In our example exercise, we perform several operations:
1. Substitution: We substitute the output of one function into another.
2. Simplification: Simplify the expressions that result from this substitution. For instance, calculating (饾憮鈭橉潙�)(鈩�) involves:
  - Find 饾憯(鈩�) first, then use this value in 饾憮.
  - Finally, solve the expression to find the result.
These steps involve basic arithmetic and understanding of function behavior.
square roots
Square roots are a mathematical operation where we find a number that, when multiplied by itself, gives the original number. For our function:
饾憯(饾懃) = 鈭氿潙�, the square root operation is fundamental:
- To find 鈭氿潙�, we determine the number y such that 饾懄 * 饾懄 = 饾懃.
When dealing with composite functions involving square roots, like (饾憯鈭橉潙�)(饾懃), understanding square roots helps us correctly simplify and solve the function combinations.

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