8. Given $g(x)=x^3$ and $h(x)=x+1$, find $(g \circ h)(-2)$.
A. 1
B. -1
C. 2
D. -2
9. If $f(x)=x^2-1$ and $g(x)=\sqrt{x}+1$. Find $(f \circ g)(x)$
A. $x+2 \sqrt{x}$
B. $x^2$
C. 2
D. $\sqrt{x}-1$.
10. If $f(x)=10 x-5$ and $g(x)=x+3$, find $(f \circ g)(x)$.
A. $10 x+25$
B. $20 x-5$
C. $10 x-25$
D. $2 x-25$.
11. Given $f(x)=3 x^2$ and $g(x)=1+2 x$ with $x \in R$, then $(f \circ g)(0)$ is
A. 3
B. 0
C. 1
D. 2.
12. If $f: X \rightarrow Y$ is such that $f(X)=Y$, then $f$ is ?
A. undefine
B. onto
C. bijective
D. one-to-one
Answer (1)
Question 8: Evaluate h ( − 2 ) = − 1 , then g ( − 1 ) = − 1 . The answer is − 1 .
Question 9: Evaluate f ( g ( x )) = f ( x + 1 ) = ( x + 1 ) 2 − 1 = x + 2 x . The answer is x + 2 x .
Question 10: Evaluate f ( g ( x )) = f ( x + 3 ) = 10 ( x + 3 ) − 5 = 10 x + 25 . The answer is 10 x + 25 .
Question 11: Evaluate g ( 0 ) = 1 , then f ( 1 ) = 3 . The answer is 3 .
Question 12: If f ( X ) = Y , then f is onto. The answer is $\boxed{onto}.
Explanation
Overview of the Problems We are given five independent math problems related to function composition and function types. We will solve each one separately.
Solving Question 8 Question 8: We are given g ( x ) = x 3 and h ( x ) = x + 1 , and we need to find ( g ∘ h ) ( − 2 ) . This means we need to find g ( h ( − 2 )) . First, we find h ( − 2 ) : h ( − 2 ) = − 2 + 1 = − 1 Now, we find g ( h ( − 2 )) = g ( − 1 ) : g ( − 1 ) = ( − 1 ) 3 = − 1 So, ( g ∘ h ) ( − 2 ) = − 1 . The answer is B.
Solving Question 9 Question 9: We are given f ( x ) = x 2 − 1 and g ( x ) = x + 1 , and we need to find ( f ∘ g ) ( x ) . This means we need to find f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( x + 1 ) = ( x + 1 ) 2 − 1 Expanding the expression, we get: ( x + 1 ) 2 − 1 = ( x + 2 x + 1 ) − 1 = x + 2 x So, ( f ∘ g ) ( x ) = x + 2 x . The answer is A.
Solving Question 10 Question 10: We are given f ( x ) = 10 x − 5 and g ( x ) = x + 3 , and we need to find ( f ∘ g ) ( x ) . This means we need to find f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( x + 3 ) = 10 ( x + 3 ) − 5 Expanding the expression, we get: 10 ( x + 3 ) − 5 = 10 x + 30 − 5 = 10 x + 25 So, ( f ∘ g ) ( x ) = 10 x + 25 . The answer is A.
Solving Question 11 Question 11: We are given f ( x ) = 3 x 2 and g ( x ) = 1 + 2 x , and we need to find ( f ∘ g ) ( 0 ) . This means we need to find f ( g ( 0 )) . First, we find g ( 0 ) : g ( 0 ) = 1 + 2 ( 0 ) = 1 Now, we find f ( g ( 0 )) = f ( 1 ) : f ( 1 ) = 3 ( 1 ) 2 = 3 So, ( f ∘ g ) ( 0 ) = 3 . The answer is A.
Solving Question 12 Question 12: We are given that f : X → Y is such that f ( X ) = Y . This means that the image of X under f is equal to Y . This is the definition of an onto function (also known as a surjective function). The answer is B.
Examples
Function composition is a fundamental concept in mathematics and has numerous applications in real life. For example, in computer graphics, transformations like scaling, rotation, and translation are often represented as functions. Combining these transformations to create complex effects involves function composition. Similarly, in economics, the cost of producing a certain number of goods can be a function of the number of goods, and the revenue generated from selling those goods can be another function. Analyzing the profit, which is revenue minus cost, involves understanding how these functions interact, often through composition.
