What are the additive and multiplicative inverses of [tex]h(x)=x-24[/tex]?
A. additive inverse: [tex]j(x)=x+24[/tex]; multiplicative inverse: [tex]k(x)=\frac{1}{x-24}[/tex]
B. additive inverse: [tex]j(x)=\frac{1}{x-24}[/tex]; multiplicative inverse: [tex]k(x)=-x+24[/tex]
C. additive inverse: [tex]j(x)=-x+24[/tex]; multiplicative inverse: [tex]k(x)=\frac{1}{x-24}[/tex]
D. additive inverse: [tex]j(x)=-x+24[/tex]; multiplicative inverse: [tex]k(x)=x+24[/tex]
Answer (2)
The additive inverse j ( x ) satisfies h ( x ) + j ( x ) = 0 , so j ( x ) = − x + 24 .
The multiplicative inverse k ( x ) satisfies $h(x) ",
Explanation
Problem Introduction We are given the function h ( x ) = x − 24 , and we want to find its additive and multiplicative inverses. Let's denote the additive inverse as j ( x ) and the multiplicative inverse as k ( x ) .
Finding the Additive Inverse The additive inverse j ( x ) is a function such that when added to h ( x ) , the result is zero. In other words, h ( x ) + j ( x ) = 0 . We can solve for j ( x ) as follows:
x − 24 + j ( x ) = 0 j ( x ) = − x + 24
Finding the Multiplicative Inverse The multiplicative inverse k ( x ) is a function such that when multiplied by h ( x ) , the result is one. In other words, $h(x) ",
Final Answer Therefore, the additive inverse is j ( x ) = − x + 24 and the multiplicative inverse is k ( x ) = x − 24 1 .
Examples
Understanding additive and multiplicative inverses is crucial in many areas of mathematics. For example, in algebra, when solving equations, we often use additive inverses to isolate variables. If we have the equation x + 5 = 10 , we add the additive inverse of 5, which is -5, to both sides to get x = 5 . Similarly, multiplicative inverses are used to divide or 'undo' multiplication. If we have 3 x = 9 , we multiply both sides by the multiplicative inverse of 3, which is 3 1 , to get x = 3 . These concepts are also fundamental in fields like cryptography and coding theory, where modular arithmetic relies heavily on finding inverses.
The additive inverse of h ( x ) = x − 24 is j ( x ) = − x + 24 , while the multiplicative inverse is k ( x ) = x − 24 1 . Therefore, the correct answer is option A.
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