Answer:
Both functions must be linear. The y-intercepts of f(x) and g(x) must be opposites.
Step-by-step explanation:
First we need to solve.
h(x) = 10 + 12 - 16
It seems tricky at first but we know a couple things. When we add the factors of h(x) together we get j(x) = 7 so we know that when we multiply 2 numbers together it should equal 10 but add up to 7.
Lets write the possible combinations of 10:
1 x 10 = 10
2 x 5 = 10
5 x 2 = 10
10 x 1 = 10
Now which combination will add or subtract to 7?
1 - 10 = 9 1 + 10 = 11
2 - 5 = -7 2 + 5 = 7 We can stop here!
2 and 5 are in our factor, so let's write it down.
(2x ) (5x ) We know there will one + and one - because the -16. If we had two + it would be positive 16, and is we had two - it would also be positive 16 because a - x - = +
Now the last number is 16. Let's find the possible combinations of 16.
1 x 16 = 16
2 x 8 = 16
4 x 4 = 16 We will stop here because we end up repeating posibilities.
Here is where we think critically. Whichever combination we choose has to be multiplied by 2 and 5 and end up equaling 12. I think 16 is too high so let's try 2 and 8.
2x * 2 = 4x 5x * 8 = 40x Now one is positive and the other is negative. Let's try each combination.
4x - 40x = -36x -4x + 40x = 36x Neither of those are 12x. So let's Try 4 and 4.
2x * 4 = 8x 5x * 4 = 20x One will be positive and the other will be negative. Let's try each combination.
8x - 20x = -12x -8x + 20x = 12x There is our combination! Remeber when you mutiply together you have to multiply the opposite factor. Here are the combinations:
(Ax + B )(Cx + D) = (Ax*Cx) + (Ax*D) + (B*Cx) + (B*D)
(Ax - B )(Cx + D) = (Ax*Cx) + (Ax*D) - (B*Cx) - (B*D)
(Ax + B )(Cx - D) = (Ax*Cx) - (Ax*D) + (B*Cx) - (B*D)
(Ax - B )(Cx - D) = (Ax*Cx) - (Ax*D) - (B*Cx) + (B*D)
So we have:
(2x + 4) (5x - 4)
Let's prove is and multiply it back out.
2x*5x - 2x*4 + 4*5x - 4*4
10 - 8 + 20
10 + 12 - 16 So we got it right!
Now let's see if they really add up to 7x.
(2x + 4) (5x - 4) which we now need to add together
2x + 4 + 5x - 4 Rearrange
2x + 5x + 4 - 4 Combine like terms
7x + 0 or 7x It works!
So our original functions are f(x) = 2x + 4 and g(x) = 5x - 4
Now to answer the question. Select 2 options.
Both functions must be linear. <em>Yes, because an equation with only an </em><em> will be a straight line, if we graph both functions they are both straight lines.</em>
Both functions must be quadratic. <em>No, because a quadratic equation is any equation that can be rearranged in standard form as </em><em> Neither f(x) or g(x) can be rearranged to fit that.</em>
Both functions must have a y-intercept of 0.<em> No, to find the y intercept we set x to 0 and solve for y.</em>
2x + 4 = y 5x - 4 = y
2(0) + 4 = y 5(0) - 4 = y
0 + 4 = y 0 - 4 = y
y = 4 y = -4
<em>Neither are 0.</em>
The rate of change of either f(x) or g(x) must be 0. <em>Let's find the rate of change for each equation. </em>
<em>We need an interval so we have to find one. Let's use 1 and 2 for x, but we have to solve for y to get the coordinate.</em>
f(x) = 2x + 4 f(x) = 2x + 4 g(x1) = 5x - 4 g(x2) = 5x - 4
x = 1 x = 2 x
y = 2(1) + 4 y = 2(2) + 4 y
y = 2 + 4 y = 4 + 4 y
y = 6 y = 8 y
( 1 , 6 ) ( 2 , 8 ) ( 1 , 1 ) ( 2 , 6 )
<em>Rate of change formula is:</em>
<em>Now we just plug in for each function.</em>
f(x) = 2x + 4 g(x) = 5x - 4
= = = 2
<em>You can see the rate of change is not 0 for either function.</em>
The y-intercepts of f(x) and g(x) must be opposites. <em>Yes, we solved for the y intercepts earlier. </em>
<em>To find the y intercept we set x to 0 and solve for y.</em>
2x + 4 = y 5x - 4 = y
2(0) + 4 = y 5(0) - 4 = y
0 + 4 = y 0 - 4 = y
y = 4 y = - 4
<em>4 and - 4 are opposites, so this statement is also true.</em>