Given that f(x) = 5x − 10 and g(x) = x + 3, solve for f(g(x)) when x = −1
f(x) = 5x − 10
g(x) = x + 3
To solve for f(g(x)), write f(x) = x + 3 and set it equal to 5x - 10 like so:
f(x + 3) = 5x - 10
Now substitute x+3 on the right side in place of x. It should look something like this:
f(x + 3) = 5(x + 3) - 10
f(x + 3) = 5x + 15 - 10
f(x + 3) = 5x + 5
Substitute the value of -1 into x.
f(x + 3) = 5(-1) + 5
f(x + 3) = -5 +5
f(x + 3) = 0
f(g(x)) = 0
Use Desmos to graph the points.
Find the slope of the line. Because the slope is positive, and we're trying to find what point intersects with the y-axis, we're going to start from point (3, 11) and end at point (1, 7).
To get from (3, 11) to (1, 7), go down 4, and to the left 2. (or left 2, and down 4 - doesn't matter which way you do it!!)
Using the slope, find your next point going down 4 and to the left 2 of (1, 7).
Your next point should be (-1, 3).
Your y-axis is the line that is vertical, so find the point at which the line you plotted intersects this axis. In this case, an easy way of calculating the intersection would be to find the point between (1, 7) and (-1, 3).
The intersection is (0, 5).
First, find out how long it would take them to read the same book. You can do this by cross multiplying.
Because it asks for the answer in minutes, change the 1 hour to 60 minutes.
100 pgs / 60 mins = 225 pgs / x mins
100x = 255(60) multiply
100x = 15300 divide 100 on both sides
x = 153
From this information, we can gather that it will take Xavier 153 minutes to read the book.
Now, calculate how long it'll take Max to read the book.
50 pgs / 60 mins = 225 pgs / x mins
50x = 255(60) multiply
50x = 15300 divide 50 on both sides
x = 308
It will take Max 308 minutes to read the book.
The question asks how many more minutes than Xavier does it take for Max to finish reading the book.
Subtract the minutes it would take Xavier to read the book from the amount of minutes it would take Max to read the book.
308 - 153 = 155
It takes 155 more minutes than Xavier for Max to finish the book.
Given that the longer leg of the right triangle is one foot less than twice the shorter leg, we can set up an equation that will help find the length of the shorter and longer leg.
a = x ft
b = 2x - 1 ft
c = 17 ft
Using the pythagorean theorem, a2 + b2 = c2, you can find the length of both the shorter and longer leg.
a2 + b2 = c2
x2 + (2x -1)2 = 172
x2 + 2x2 + 1 = 289
3x2 / 3 = 288 / 3
x2 = 96
sqrt(x2) = sqrt(96)
x = 9.8
The length of the shorter leg is 9.8 ft.
To find the length of the longer leg, substitute 9.8 for x in 2x - 1.
2(9.8) - 1 =
19.6 - 1
The length of the longer leg is 18.6 ft.
To find the area of a triangle, multiply the base by the height, and then divide by 2.
area = ( 9.8 x 18.6 ) / 2
area = 182.28 / 2
area = 91.12 ft
The area of the triangle is 91.12 ft.
To solve this problem, I went to Youtube. If you want to follow along and do it, here's the link to the video I used, but if you want an explanation without watching a video, I will do my best to explain below.
Plot the points given. I labeled them KL in the graph below.
As you can see, I have made a lot of side notes. It may be hard to understand at first, but bare with me.
I drew a straight line across from K, and then a straight line down from L, connecting the both into a triangle.
The point I labeled at (?,?) is the point we're trying to find. It's only an APPROXIMATE spot!!!! This is just a visual for your understanding.
To divide the segment into the 5:3 ratio, we must conclude that there's some point that divides x into the ratio 5:3 and there's some point that divides y into the ratio 5:3. (I represented this by putting 5x and 3x, and 5y and 3y.)
Then, we need to think about what the total change in x is from point K (-4, -3) to point L (5, 3). Doing this, we find the distance between the x on each point. So ask yourself what the distance is from -4 to 5.
It is going to be 9. (I represented this on the image, where delta = 9)
Next, we do the same for the distance between y on each point. Ask yourself what the distance is from -3 to 3.
It's going to be 6. (I also represented this on the image too, where delta = 6.)
Now that you understand everything on the graph, you can use the information given to find the point that divides the segment from each point.
Form two equations with x and y from the graph, and then solve. This is not the last step yet, sadly.
5x + 3x = 9
8x = 9
x = 9/8
3y + 5y = 6
8y = 6
y = 6/8
y = 3/4
Using the point (-4, -3), we can use the x and y values to finally determine the point that divides the segment from each point.
Let's label (?,?) as P.
P ( -4 + (5 x (9/8)), -3 + (5 x (3/4)) )
Solve for for point P.
P ( -4 + (5 x (9/8)), -3 + (5 x (3/4)) )
P ( -4 + 5.525, -3 + 3.75 )
P ( 1.625, 0.75 )
OR you can flip it into a fraction.
P ( 13/8 , 3/4 )
The point that divides the segment from each point can be represented at ( 13/8 , 3/4 ).
List the factors of 216.
- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Since his ID code is only three-digits, we can eliminate the numbers above 9, which leaves us to:
- 1, 2, 3, 4, 6, 8, 9
Use these numbers to determine what 3 numbers add to equal 19 and multiply to equal 216.
The numbers in Angel's ID code include: 4, 6, and 9.
List the possible three-digit outcomes of 4, 6, 9 (hint - since there are three digits, there are 6 outcomes):
The last bit of information also tells us that the integer code is as large as possible. Picking from the list of possible three-digit outcomes, ask yourself, which is the largest possible?
Angelo's ID code is 9-6-4.
If you have any questions, don't be afraid to ask! I hope I explained well enough.
(x + y)/2 = 5
sqrt(xy) = 3
sqrt(xy)2=32 Square both sides to eliminate the square root on the left side.
xy = 9
(xy)/x = 9/x Divide x on both sides.
y = 9/x
Now that you know y, you can solve for x by using the first equation given.
(x + y)/2 = 5
(x+(9/x))/2 = 5 Substitute 9/x into the value of y.
(x+(9/x))/ 2 ⋅ 2 = 5 ⋅ 2 Multiply 2 on both sides to get rid of the denominator on the left side.
x+(9/x) = 10
x+(9/x) - x = 10 - x Subtract x on both sides.
9/x = 10 - x
(9/x) ⋅ x = (10 - x) ⋅ x Multiply x on both sides to get rid of the denominator once again.
9 = 10x - x2
9 - 9 = 10x -x2 -9 Subtract 9 to move everything onto one side.
-x2 +10x - 9 = 0 Rewrite.
-1 (-x2 +10x - 9 = 0) From here, we must factor to get the values of x. But first we must mulitply the equation by -1.
x2 - 10x + 9 = 0 Factor.
(x+10) (x-1) = 0 Now set each of the parenthesis to equal 0.
x + 10 = 0
x = -10
x - 1 = 0
x = 1