college algebra // midterm review

To graph f(x) = x2 − 4, we plot the following points.

(x, x^2 - 4) -just imput f(x) into the y spot

Consider the following functions. *f(x) = 4x + 5, g(x) = 3x − 1*

-should have no domain issues for any of the problems -plug in what the problem is asking

Evaluate the piecewise defined function at the indicated values. f(x) = *x^2 + 6x* if x ≤ −1 *x* if −1 < x ≤ 1 *−1* if x > 1

find out which category the f(n) fits into and then use that number for x and find out the answer

Solve the system, or show that it has no solution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.) *10x + 4y = 16* *15x + 6y = 24*

-can multiply some of the sides so that you can cancel out an x or y -if you get 0=0, then you need to find y. do that by solving the first or second equation but have it equaled to y -type answer as (x, n-n/2x)

Use the Inverse Function Property to determine whether f and g are inverses of each other. *f(x) = 4x + 7; g(x) = (x − 7)/4

-do (fog) and (gof) and if both answers =x they are inverses

Use *f(x) = 4x − 3* and *g(x) = 5 − x^2* to evaluate the expression. (f o g) (-2) (g o f) (-2)

-first you have to find what g is -afterwards you can put in what f(-2) is -find f(-2) first and then find g(-2)

A rectangular garden is 9 ft longer than it is wide. Its area is 442 ft^2. What are its dimensions? *width-* *length*

-know that the formula will look like w(w+9)=442ft^2 -factor it out then use the positive answer for the width and then add the extra 9 for the length

The graph of a function f is given. Use the graph to estimate the following. *local max and mins* *increasing and decreasing*

-make sure to pay attention to the numbers -go from left to right when finding mins and maxs -increasing and decreasing will start go from neg to pos also

Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) *|2x − 5| = 9*

-make two equations, one with a negative answer and the other with a positive -figure out what x is for both equations

Find all real solutions of the equation. (Enter your answers as a comma-separated list.) *x^3 − 2x^2 − 7x + 14 = 0*

-separate the problem into factoring: x^3 and 2x^2 will be the outside numbers and the 7x and 14 will be used as the factoring in parenthesis -one answer will be in square root so make sure to write the positive and negative answer

Find all real solutions of the equation. (Enter your answers as a comma-separated list.) *√(x + 42)=x*

-square the square root to cancel it out -afterwards you can factor the problem to find out what x is -x has to be the positive answer

Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) *x^4 − 8x^2 + 7 = 0*

-this is one you need to change the x to w; x^4=w^2 -after factoring the w, change the w=(number) to x^2=(number); then get rid of the square in x -will have 4 answers because you need the +/-

Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) *|x − 5| = −5*

-this one would be no solution -an absolute value question cannot start with a negative answer

Find all solutions of the equation and express them in the form a + bi. (Enter your answers as a comma-separated list. Simplify your answer completely.) *x^2 + 3x + 3 = 0*

-use *i* with quadratic equation because you will get a negative square root -make sure to use the +/- sign and for answer put *i* after the fraction

Solve the system, or show that it has no solution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.) *3x + 2y = −10* *−x − 2y = 14*

-use elimination and find both x and y

Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) *x^2 − 10x + 1 = 0*

-use quadratic formula -can factor if you know the factors

Find f + g,f − g, fg, and f/g and their domains. f(x) = x, g(x) = 2x

-should only have domain issues in the (f/g) -look at notes for all the answers

Find the domain of the function. (Enter your answer using interval notation.) *f(t) = ∛(t − 1)*

domain would be (- infinity, + infinity) because an odd square root has no domain issues

Find f(a), f(a + h),and the difference quotient f(a + h) − f(a)/h where h ≠ 0. *f(x) = 6 − 4x*

plug in 6-4x where it needs to go in the equation

Solve the system of linear equations. (If the system is dependent, express your answer in terms of t, where x = x(t), y = y(t), and z = t. If the system is inconsistent, enter INCONSISTENT.) *4x − 3y + z = −46* *−2x + y − 3z = 10* *x − y + 2z = −6*

use matrix on calendar then type in answers

*inverse graph question* does f have an inverse? If so, find the following. (If an answer does not exist, enter DNE.) f^-1(1) f^-1(6)

-will be able to be an inverse if passes horiz line test -to find inverse write (1, ?) and (6, ?). go to graph and find those xs on the graph where they intersect with a y. whatever y point they intersect with is the answer

Find the domain of the function. (Enter your answer using interval notation.) *g(x) = 6 + x / 9 − x*

-will have domain issues -4-x cant = 0 -x has to be greater than or equal to -5 domain would be [-5, 4) U (4, infinity)

A boat on a river travels downstream between two points, 80 mi apart, in 1 h. The return trip against the current takes 2 1/2 h. What is the boat's speed (in still water)? boat speed= 56mph river current=24 mph

-you have to do (b+c)*time=miles to find the downstream use (b-c)*time=miles to find the upstream -use elimination to find the boat speed for the two previous equations -to find the current use b+c=miles

Find the inverse function of f. *f(x) = 3x + 9*

change f(x) to y, then change y into x, divide by y and then find what the y= -type in the answer you got from y=

Find the inverse function of f. *f(x) = x − 6 / x + 6*

change x to y, multiply (y+6) to each side, after getting all the ys on one side, divide by the y(n) n in parethesis to get something like y= -6-6x/x-1

Let S = −5, −1, 0, 2/3, 5/6, 1, √6, 6, 7. Determine which elements of S satisfy the inequality. (Enter your answers as a comma-separated list.) *−2 + 3x ≥ 1/3*

get x by itsself, after finding x alone figure out which S= are the answer

Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. *y = f(x − 6) + 2* *y = f(x + 4) − 1*

shift 6 units to the right and 2 units upward shift 4 units to the left and 1 unit downward

A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. f(x) = |x|; shift 8 units to the left and shift downward 4 units

|x+8| -4