1. Which of the following maps is not a function?
A. [tex]$y=x^3-1$[/tex]
B. [tex]$y^2-x^2=6$[/tex]
C. [tex]$x^{3 / 2}=y+1$[/tex]
D. [tex]$y-x^4=20$[/tex].
2. Find [tex]$f(1)$[/tex] if [tex]$f(x)=2 x^2-5 x+3$[/tex].
A. 1
B. 4
C. -1
D. 0 .
3. Find the roots of [tex]$h(y)=|y+4|-1$[/tex].
A.3,5
B. -3,5
C. -3,-5
D. 3,-5
E. 4,5
4. Find the roots [tex]$f(x)=\frac{3 x^2+2 x-5}{x+2}$[/tex].
A. [tex]$x=5$[/tex] or -3
B. [tex]$x=-5$[/tex] or -3
C. 1 or [tex]$-\frac{5}{3}$[/tex]
D. 3 or 5 .
5. Find the zeros of [tex]$f(w)=|w+3|-7$[/tex]
A. 3, 7
B. 7, 6
C . 4, - 10
D 9, 3 .
6. One of the following is a root of [tex]$g(t)=9 t^3-18 t^2+6 t$[/tex].
A. [tex]$\frac{\sqrt{3}}{3}$[/tex]
B. 0
C. [tex]$\frac{\sqrt{3}}{2}$[/tex]
D. [tex]$\frac{\sqrt{2}}{3}$[/tex].
7. The multiplicity of 0 as a root of [tex]$p(x)=2 x^4-2 x^3-12 x^2$[/tex] is...
A. 1
B. 2
C. 3
D. 4 .
8. Given [tex]$g(x)=x^3$[/tex] and [tex]$h(x)=x+1$[/tex], find [tex]$(g \circ h)(-2)$[/tex].
A. 1
B. -1
C. 2
D. -2
9. If [tex]$f(x)=x^2-1$[/tex] and [tex]$g(x)=\sqrt{x}+1$[/tex]. Find [tex]$(f \circ g)(x)$[/tex]
A. [tex]$x+2 \sqrt{x}$[/tex]
B. [tex]$x^2$[/tex]
C. 2
D. [tex]$\sqrt{x}-1$[/tex].
10. If [tex]$f(x)=10 x-5$[/tex] and [tex]$g(x)=x+3$[/tex], find [tex]$(f \circ g)(x)$[/tex].
A. [tex]$10 x+25$[/tex]
B. [tex]$20 x-5$[/tex])
C. [tex]$10 x-25$[/tex]
D. [tex]$2 x-25$[/tex].

The answers to the set of questions cover key concepts like functions, evaluating equations, and finding roots. The first question identifies a non-function from given options, and subsequent questions involve evaluations and roots of various functions. Overall, the detailed solutions help clarify the steps needed to determine the answers to each question.
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Answered by Anonymous | 2025-07-08

Question 1: A function must have a unique y-value for each x-value. y 2 − x 2 = 6 is not a function.
Question 2: Evaluate f ( 1 ) for f ( x ) = 2 x 2 − 5 x + 3 , which gives 0 ​ .
Question 3: Solve ∣ y + 4∣ − 1 = 0 , which gives roots − 3 , − 5 ​ .
Question 4: Find the roots of x + 2 3 x 2 + 2 x − 5 ​ , which are 1 , − 3 5 ​ ​ .
Question 5: Solve ∣ w + 3∣ − 7 = 0 , which gives zeros 4 , − 10 ​ .
Question 6: Find a root of 9 t 3 − 18 t 2 + 6 t = 0 , which is 0 ​ .
Question 7: The multiplicity of 0 as a root of 2 x 4 − 2 x 3 − 12 x 2 is 2 ​ .
Question 8: Given g ( x ) = x 3 and h ( x ) = x + 1 , ( g ∘ h ) ( − 2 ) = − 1 ​ .
Question 9: Given f ( x ) = x 2 − 1 and g ( x ) = x ​ + 1 , ( f ∘ g ) ( x ) = x + 2 x ​ ​ .
Question 10: Given f ( x ) = 10 x − 5 and g ( x ) = x + 3 , ( f ∘ g ) ( x ) = 10 x + 25 ​ .

Explanation

Question 1: Identifying Non-Functions For a mapping to be a function, each input (x-value) must correspond to exactly one output (y-value). In option B, y 2 − x 2 = 6 , we can express y as y = ± x 2 + 6 ​ . This means that for a single value of x , there are two possible values of y (one positive and one negative), so it is not a function.

Question 2: Evaluating Functions To find f ( 1 ) for f ( x ) = 2 x 2 − 5 x + 3 , substitute x = 1 into the expression: f ( 1 ) = 2 ( 1 ) 2 − 5 ( 1 ) + 3 = 2 − 5 + 3 = 0 .

Question 3: Finding Roots of Absolute Value Functions To find the roots of h ( y ) = ∣ y + 4∣ − 1 , set h ( y ) = 0 : ∣ y + 4∣ − 1 = 0 ⇒ ∣ y + 4∣ = 1 . This gives two cases: y + 4 = 1 or y + 4 = − 1 . Solving these gives y = − 3 or y = − 5 .

Question 4: Finding Roots of Rational Functions To find the roots of f ( x ) = x + 2 3 x 2 + 2 x − 5 ​ , we need to solve 3 x 2 + 2 x − 5 = 0 . Factoring gives ( 3 x + 5 ) ( x − 1 ) = 0 , so x = 1 or x = − 3 5 ​ .

Question 5: Finding Zeros of Absolute Value Functions To find the zeros of f ( w ) = ∣ w + 3∣ − 7 , set f ( w ) = 0 : ∣ w + 3∣ − 7 = 0 ⇒ ∣ w + 3∣ = 7 . This gives two cases: w + 3 = 7 or w + 3 = − 7 . Solving these gives w = 4 or w = − 10 .

Question 6: Finding Roots of Polynomials To find a root of g ( t ) = 9 t 3 − 18 t 2 + 6 t , we solve 9 t 3 − 18 t 2 + 6 t = 0 . Factoring out 3 t gives 3 t ( 3 t 2 − 6 t + 2 ) = 0 . Thus, t = 0 is one root.

Question 7: Determining Multiplicity of Roots To find the multiplicity of 0 as a root of p ( x ) = 2 x 4 − 2 x 3 − 12 x 2 , factor the polynomial: p ( x ) = 2 x 2 ( x 2 − x − 6 ) = 2 x 2 ( x − 3 ) ( x + 2 ) . The factor x 2 indicates that 0 is a root with multiplicity 2.

Question 8: Composite Functions Given g ( x ) = x 3 and h ( x ) = x + 1 , we find ( g ∘ h ) ( − 2 ) = g ( h ( − 2 )) . First, h ( − 2 ) = − 2 + 1 = − 1 . Then, g ( − 1 ) = ( − 1 ) 3 = − 1 .

Question 9: Composite Functions Given f ( x ) = x 2 − 1 and g ( x ) = x ​ + 1 , we find ( f ∘ g ) ( x ) = f ( g ( x )) = f ( x ​ + 1 ) = ( x ​ + 1 ) 2 − 1 = ( x + 2 x ​ + 1 ) − 1 = x + 2 x ​ .

Question 10: Composite Functions Given f ( x ) = 10 x − 5 and g ( x ) = x + 3 , we find ( f ∘ g ) ( x ) = f ( g ( x )) = f ( x + 3 ) = 10 ( x + 3 ) − 5 = 10 x + 30 − 5 = 10 x + 25 .


Examples
Understanding functions and their properties is crucial in many real-world applications. For example, composite functions can model sequential processes, such as a discount applied after a tax. Finding roots helps in optimization problems, like determining the break-even point for a business. Analyzing the multiplicity of roots is essential in understanding the behavior of systems near equilibrium, such as in chemical reactions or population dynamics. These concepts provide a foundation for advanced mathematical modeling and problem-solving.

Answered by GinnyAnswer | 2025-07-08