Daniel's savings account balance is 20 times the amount of Henrys savings account balance. The total amount of money contained i
2 answers:
Answer: Daniel has $22 in his savings account balance.
Step-by-step explanation:
Let Daniel's savings account balance be x
and Henry's savings account balance be y, then as given Daniel's savings account balnce is 20 times that of Henry's, We get ...........(1)
And also total amount of both savings account is $462 so we get ......(2)
Now by substituting values of x from (1) into (2) we get,
And
So, Daniel's savings account balance is $440
and Henry's savings account balance is $22
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Answers:
a = 2 and b = -4
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Explanation:
Let's define the three helper functions
- f(x) = ax^2 - b
- h(x) = 6
- j(x) = 5ax+b
which are drawn from the piecewise function. The g(x) function will change depending on what the input is.
- If x < 1, then g(x) = f(x).
- If x = 1, then g(x) = h(x)
- If x > 1, then g(x) = j(x)
Since we want g(x) to be continuous at x = 1, this must mean the three functions f(x), h(x), j(x) must have the same output value when the input is x = 1.
Because h(x) = 6 is a constant function, the output is always 6 regardless of the input. Therefore, we want f(x) and j(x) to have 6 as their output when x = 1. Or else, the pieces won't connect.
Plug x = 1 into the f(x) function to get
f(x) = ax^2 - b
f(1) = a(1)^2 - b
f(1) = a - b
Set this equal to the desired output of 6 and we end up with the equation a-b = 6. Solving for 'a' leads to a = b+6.
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We'll use the same idea for j(x)
j(x) = 5ax + b
j(1) = 5a(1) + b
j(1) = 5a + b
5a+b = 6
5(b+6) + b = 6 ... plug in a = b+6; solve for b
5b+30+b = 6
6b+30 = 6
6b = 6-30
6b = -24
b = -24/6
b = -4
Which then leads to,
a = b+6
a = -4+6
a = 2
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Since a = 2 and b = -4, we go from this
to this
Meaning
f(x) = 2x^2+4 and j(x) = 10x-4
You should find that plugging x = 1 into each of those two functions leads to 6 as the output.
The graph is shown below. Note the red graph f(x) is only drawn when x < 1. Similarly, j(x) is only drawn when x > 1. The orange point represents h(x) which only happens when x = 1. So as the name implies, the piecewise function g(x) is composed of pieces of the three functions f(x), h(x), j(x).
