Precalculus Final Review

Fundamentals

100

Evaluate 

5^23/5^21

25

200

Factor 

2x^2-5x-12

(2x-3)(x+4)

300

Find all the real solutions of the equation 

x^4-3x^2+2=0

+-1, +-sqrt(2)

400

Simplify 

((3x^(3/2)y^3)/(x^2y^(-1/2)))^(-2)

x/(9y^7)

500

Simplify 

((x^2+2x-3)/(x^2+8x+16))/((x-1)/(3x+12)

(3(x+3))/(x+4)

Functions (Does not include composition or inverse of functions)

100

Find the domain of 

1+ sqrt(x)

[0,infty)

200

Find the net change of f(x) between x=1 and x=4                               

f(x)=x^2-2x. 

Net change is 9. 

300

Find the average rate of change of f(x) between x=1 and x=4. 

f(x)=x^2-2x

Average rate of change is 3. 

400

Supposed the graph of f is given. Describe how the graph of the following function can be obtained from the graph of f. 

y=-f(x-2)-2

Function f is reflected over the x-axis, shifted right 2 units, and shifted down 2 units. 

500

Determine whether f is even,  odd, or neither. 

f(x)=2x^5-3x^2+2

Neither 

Combining functions & Inverse functions

100

Find g(f(-2))

f(x) = 2x-3

g(x)=4-x^2

-45

200

Find f + g. 

f(x) = x^2-3x+2

g(x)=4-3x

x^2-6x+6

300

Find the inverse function of 

f(x)=1/(x-2)

f^-1 (x)=1/x+2

400

Find f(g(x))

f(x) = x^2-3x+2

g(x)=4-3x

9x^2-15x+6

500

Find the inverse function of 

f(x)=(x-3)/(2x+5)

f^-1 (x)= - (5x+3)/(2x-1)

Polynomial & Rational Functions

100

Describe the end behaviour of the polynomial function P. 

P(x)=-(x-1)^2(x-4)(x+2)^2

As x-> negative infinity, P(x)-> positive infinity. 

As x-> positive infinity, P(x)-> negative infinity.

200

Find the maximum or minimum value of the quadratic function 

f(x)=-x^2+3x-1

Maximum of 5/2 when x=3/2 

300

Determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. 

P(x)=3x^7-x^5+5x^4+x^3+8

2 or 0 positive. 

3 or 1 negative. 

400

What is the domain and range of r(x)? 

r(x)=(x^2+5x-14)/(x-2)

D: (-infty, 2)cup(2,infty)

R: (-infty, 9)cup(9,infty)

500

Find the x-intercept, y-intercept, domain, and range of the rational function r(x). 

r(x)=(x^2+3x-18)/(x^2-8x+15)

x-intercept: (-6, 0) 

y-intercept: (0, -6/5) 

D: (-infty, 3)cup(3,5)cup(5,infty) 

R: (-infty, -9/2)cup(-9/2,1)cup(1,infty)

Exponential and Logarithmic Functions

100

Evaluate the expression without a calculator. 

log25+log4

2 

200

Expand the logarithmic expression. 

lnsqrt((x^2-1)/(x^2+1)

1/2[ln(x+1)+ln(x-1)-ln(x^2+1)]

300

Solve the equation. 

3^(2x-7)=27

x=5

400

Combine into a single logarithm. 

log(x-2)+log(x+2)-1/2log(x^2+4)

log((x^2-4)/(sqrt(x^2+4)))

500

Solve the equation. 

log_3(x-8)+log_3x=2

x=9