Download f(x)

Name: __________________________________ Date:________________ Block:______ PreCalc: Mrs. Knox
MIDTERM
Midterm over Spring Break Due Monday April 13th. If not handed in on the 13th there will be 20
points automatically deducted. Complete work in PENCIL! SHOW ALL WORK FOR FULL CREDIT!
If necessary show work on separate piece of paper but write all answers on the test.
1 1 6
1. −22, −7.5, −1, − 3 , 2 , 3 , 8, √2, 𝜋
Determine which numbers are:
a.) Natural numbers:____________________________________________
b.) Integers:__________________________________________________
c.) Rational numbers:___________________________________________
d.) Irrational numbers:__________________________________________
2. Evaluate the expression.
b.) 4 − 2𝑥 2 when 𝑥 = −2
a.) −2 − |−4|
3. Simplify each expression. Rewrite each expression with positive exponents.
a.) (3𝑎2 𝑏 5 )(−4𝑎4 𝑏 −1 )
b.)
16𝑎5 𝑎−2
2𝑎2 𝑏3
4. Simplify. Write in standard form.
a.) (5𝑥 3 − 7𝑥 2 − 3) + (𝑥 3 + 2𝑥 2 − 𝑥 + 8)
b.) (3𝑥 − 2)2
5. Factor
a.) 𝑛2 − 6𝑛 + 8
b.) 𝑦 2 + 2𝑦 − 15
c.) 2𝑥 2 − 7𝑥 + 3
6. Perform the operation and simplify. (6 pts each)
a.)
b.)
𝑥 2 +4𝑥+3
𝑥 2 +5𝑥+6
∙
3𝑥 2 +4𝑥+1
𝑥 2 −4
𝑥 2 −3𝑥−10
𝑥 2 +𝑥
÷
𝑥+1
𝑥 2 +8𝑥+12
7. Write the set of numbers in set-builder and interval notation, if possible.
𝑥 ≤ −16 or 𝑥 > 5
−2x − 1 if 𝑥 < 2
𝑥 2 + 1 if 2 ≤ 𝑥 ≤ 5
8. If f(x) = {
5x − 3 if x > 5
, Evaluate the following
a. 𝑓(−2)
9. State the domain of f(x) =
b. 𝑓(10)
𝟐𝐭−𝟔
𝒕𝟐 +𝟔𝒕+𝟗
10. State the domain and range of the functions shown.
a. Domain:
b. Domain:
Range:
Range:
11. Find the x and y intercepts algebraically. Check by looking at the graph.
12. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin. Show
algebraically.
13. GRAPHING CALCULATOR Graph the function. Analyze the graph to determine whether each
function is even, odd, or neither. Confirm algebraically. If odd or even, describe the symmetry of the
graph of the function.
a. 𝑓(𝑥) = 𝑥 3 − 2𝑥
14. Use the graph of each function to estimate intervals to the nearest 0.5 unit on which the function is
increasing, decreasing, or constant.
a.
b.
Name the parent function. Then describe the transformation.
15. g(x) = √𝑥 + 5 + 3
16. Find the following for the given functions. State the domain of each new function.
𝑓(𝑥) = 𝑥 2 + 4𝑥 and 𝑔(𝑥) = 3𝑥 − 5
a. (𝑓 + 𝑔)(𝑥)
𝑔
b. (𝑓 )(𝑥)
17. Find the following for 𝑓(𝑥) = 𝑥 2 + 1 and 𝑔(𝑥) = 𝑥 − 4
a. 𝑔(𝑓(𝑥))
18. Find 𝒇(𝒈(𝒙)) and state the domain.
1
𝑎. 𝑓(𝑥) = 𝑥+1 and 𝑔(𝑥) = 𝑥 2 − 9
19. Find two functions 𝒇 and 𝒈 such that 𝒉(𝒙) = 𝒇(𝒈(𝒙))
ℎ(𝑥) = √𝑥 3 − 4
𝑓(𝑥) =
𝑔(𝑥) =
20. Find the inverse function and state any restrictions on the domain.
c. 𝑔(𝑥) = √𝑥 − 4
d. 𝑓(𝑥) = 2𝑥 + 7
21. Show algebraically that f and g are inverse functions.
6
6
e. 𝑓(𝑥) = 𝑥 + 4 and 𝑔(𝑥) = 𝑥−4
𝟏
22. Graph and analyze f(x) = 𝟐 𝒙𝟒 . Describe the domain, range, intercepts, end behavior, and where the function is
increasing or decreasing.
x
–3
–2
–1
0
1
2
3
f(x)
Domain:
Range:
Intercept:
End behavior:
Increasing/Decreasing:
23. Graph and analyze f(x) = −𝒙𝟕 . Describe the domain, range, intercepts, end behavior, and where the function is
increasing or decreasing.
x
–3
–2
–1
0
1
2
3
f(x)
Domain:
Range:
Intercept:
End behavior:
Increasing/Decreasing:
Solve each equation. Check for extraneous solutions.
3
24. √2𝑥 + 7 =3
25. 𝑥 + 1 = √7𝑥 + 15
26. Describe the end behavior of 𝑓(𝑥) = −3𝑥 2 − 2𝑥 7 + 4𝑥 4 using the leading term test.
27. What is the greatest possible number of real zeros and turning points of 𝑓(𝑥) = 3𝑥 6 − 10𝑥 4 − 15𝑥 2?
State the number of possible real zeros and turning points of each function.
Then determine all of the real zeros by factoring. State any even or odd multiplicity.
28. f(x) = 𝑥 3 − 5𝑥 2 + 6𝑥
29. f(x) = 𝑥 4 – 9𝑥 2 + 18
30. For f(x) = x(𝑥 − 1)2 (2x + 3), (a) apply the leading term test, (b) determine the zeros and state the multiplicity of any
repeated zeros, (c) find a few additional points, and then (d) graph the function.
a)
d)
b)
c)
Interval
x-value
yvalue
Coordinate
(x,y)
**Remember to go back and check your answers. Please circle your answers. Do not leave anything blank. Did you use your book and notebook to help you????**