Precalculus Test

HUDSON COUTNY COM𝑀UNITY COLLEGE MAT 110_ final exam

1. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛:

𝑓(𝑥) = √1 − 𝑥2

2. Determine all vertical asymptotes:

𝑓(𝑥) = 8 − 4𝑥

𝑥2 − 4

3. 𝑆𝑖𝑚𝑝𝑙𝑓𝑦:Expand the expression,

ln ( 𝑥2√𝑥 − 1

𝑦4(2𝑥 + 5)3 )

4. 𝐿𝑖𝑛𝑒𝑎𝑟𝑖𝑧𝑒 𝑡ℎ𝑒 𝑡𝑟𝑖𝑔𝑜𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛: 𝑠𝑖𝑛24𝑥𝑐𝑜𝑠25𝑥 − 𝑐𝑜𝑠24𝑥𝑠𝑖𝑛25𝑥

5. Solve the equation:

𝑙𝑛𝑥 + ln (2𝑥 + 1) = 5

6. Determine all horizontal asymptotes

𝑦 = 𝑥 − 1

𝑥 + 1

7. Determine all asymptotes:

𝑦 = 𝑥

𝑥 − 1

8. Determine all vertical asymptotes:

𝑦 = 2𝑥 + 1

𝑥 − 9

9. Find the domain of the function:

𝑦 = ln (𝑥 + 1) − 𝑙𝑛(𝑥 + 2)

10. Determine the possible values of the base b, if the function 𝑦 = 2𝑏−𝑥, has the –x-axis as a horizontal asymptote

11. Determine the real zero of the function 𝑦 = ( 𝑥2 − 2)(𝑥2 − 4𝑥 − 12)(𝑥2 + 1)

12. Given 𝑓(𝑥) = √𝑥2 + 𝑥 − 6 , 𝑔(𝑥) = −2𝑙𝑜𝑔3(𝑥 + 1), 𝑤(𝑥) = 𝑡𝑎𝑛𝑥, 𝑎𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓ℎ 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒

x ℎ(𝑥) -2 -3

0 1

3 -2

4 0

Find: 𝑎). 𝑤 ( 5𝜋

3 ) ; b). 𝑔 ( √9

3 − 1) ;c). (𝑓 + 𝑔)(2) ;d). 𝑔 (3𝑥 − 1)

13. Determine all vertical asymptotes:

𝑓(𝑥) = 𝑙𝑛(𝑥 − 4)

14. Find the domain of the function:

𝑦 = 𝑥+2

𝑥2+10

15. 𝑓(𝑥) = 𝑥2 + 𝑥 − 6 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥−3

𝑥−3 ,

𝐹𝑖𝑛𝑑: 𝑎. 𝑓(𝑥 + ℎ) − 𝑓(𝑥)

ℎ , ℎ ≠ 0, 𝑏.

𝑔(𝑥 + ℎ) − 𝑔(𝑥)

ℎ

16. Solve the equation: 2 𝑠𝑖𝑛2𝑥 − 𝑠𝑖𝑛𝑥 = 0

17. 52𝑥−1 = 7𝑥+3

18. 𝑙𝑜𝑔3(𝑥 + 2) + 𝑙𝑜𝑔3(𝑥 − 2) = 2

19. 𝑓(𝑥) = 3sin(2𝑥 + 𝜋

3 ), Find period, amplitude and phase shift

20. Find the equation of each line in slope-intercept form. Then b) determine

whether the lines are parallel, perpendicular or neither.

𝐿1: (0,1) , ( 5, 9) 𝐿2: ( 0,3), ( 4, 1)

22. Write as a single logarithm 𝑙𝑜𝑔2𝑎

3 +

𝑙𝑜𝑔2𝑏

3 +

𝑙𝑜𝑔2𝑐

3 =