Need help on 6. 8th grade math
1 answer:
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Answer:
6
Step-by-step explanation:
If the question were how many distinct four digit number contain the digits 1,2,3, and 4 without repetition we could use the variation without repetition formula for 4 elements.
The first digit must be any of this four digits, so, we have 4 possibilities.
For the second digit, we already used one digit, so, we only have 3 possibilities. The total possibilities of picking this first two numbers are 4 * 3.
For the third digit, we already used two digits, so, we have 2 possibilities. The total possibilities thus far are then 4 * 3 * 2.
And for the final digit we don't have a choice, its the only one left. So we have 4 * 3 * 2 * 1 possibilities for a four digit number formed by the digits 1,2,3, and 4 without repetition .
This number its the factorial of 4:
.
If the digit 2 is always immediately followed by the digit 3, no, we don't have 4 choices, cause, our elements are:
the digit 1 , the duple 2 3 , and the digit 4
So, for the first position, we must choose one of this elements. Lets say 1.
1 _ _ _.
Then, we can choose the duple 23 for the second and third position:
1 2 3 _
Finally we only got one choice, the digit 4:
1 2 3 4
So, as we can see. This is the same that choosing three elements without repetition. The possibilities will be:
.
Now, we could: ask which are this possibilities?. They are:
1 2 3 4
1 4 2 3
2 3 1 4
2 3 4 1
4 1 2 3
4 2 3 1
The transformation of a function may involve any change. The function f(x) is vertically stretched and shifted 5 units upwards to form h(x).
<h3>How does the transformation of a function happen?</h3>The transformation of a function may involve any change.
Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs) etc.
If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:
Horizontal shift (also called phase shift):
- Left shift by c units, y=f(x+c) (same output, but c units earlier)
- Right shift by c units, y=f(x-c)(same output, but c units late)
Vertical shift
- Up by d units: y = f(x) + d
- Down by d units: y = f(x) - d
Stretching:
- Vertical stretch by a factor k: y = k \times f(x)
- Horizontal stretch by a factor k: y = f(\dfrac{x}{k})
The function f(x)=x^(1/3) is transformed to form the function of h(x)=(2x)^(1/3)+5. Therefore, the transformation made to the function is,
Vertically stretched by a factor of 2^(1/3) ⇒ 2^(1/3) × x^(1/3) = (2x)^(1/3)
Up by 5 units ⇒ (2x)^(1/3) + 5
Hence, the function f(x) is vertically stretched and shifted 5 units upwards to form h(x).
Learn more about Transforming functions:
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