Let the number of years Brad has been on the soccer team be represented by x, and let y represent the number of years Scott has been on the same team.
From the information given, we formulate the equations:
5y - 2 = x -----(1)
x + y = 10 -----(2)
Now, substituting equation (1) into equation (2):
5y - 2 + y = 10
6y - 2 = 10
6y = 12
y = 2
Next, substituting this value back to find x:
x = 5×2 - 2
x = 8
So, Brad has spent a total of 8 years in the team.
Response:
Sampling with replacement signifies that a marble is selected, its color is noted, replaced, and the box is shaken before another selection is made.
There are
marbles in total. The probability of selecting the first blue marble is
, then when Maya places this marble back, the chance of picking the second blue marble remains constant. By applying the product rule, the probability of drawing 2 blue marbles consecutively is
In certain cases, a function necessitates multiple formulas to achieve the desired outcome. An example is the absolute value function \displaystyle f\left(x\right)=|x|f(x)=∣x∣. This function applies to all real numbers and yields results that are non-negative, defining absolute value as the magnitude or modulus of a real number regardless of its sign. It indicates the distance from zero on the number line, requiring all outputs to be zero or greater.
<pwhen inputting="" a="" non-negative="" value="" the="" output="" remains="" unchanged:="">
\displaystyle f\left(x\right)=x\text{ if }x\ge 0f(x)=x if x≥0
<pwhen inputting="" a="" negative="" value="" the="" output="" is="" inverse:="">
\displaystyle f\left(x\right)=-x\text{ if }x<0f(x)=−x if x<0
Due to the need for two distinct operations, the absolute value function qualifies as a piecewise function: a function defined by several formulas for different sections of its domain.
Piecewise functions help describe scenarios where rules or relationships alter as the input crosses specific "boundaries." Business contexts often demonstrate this, such as when the cost per unit of an item decreases past a certain order quantity. The concept of tax brackets also illustrates piecewise functions. For instance, in a basic tax system where earnings up to $10,000 face a 10% tax, additional income incurs a 20% tax rate. Thus, the total tax on an income S would be 0.1S when \displaystyle {S}\leS≤ $10,000 and 1000 + 0.2 (S – $10,000) when S > $10,000.
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This explanation follows: The query lacks completeness; please refer to the complete question in the provided attachment. The curve described is represented by the quadratic equation, y = x². If the curve moves downwards by 2 units, the equation for the new curve becomes y = x² - 2. Shifting the curve 2 units to the left results in the equation y = (x + 2)². Conversely, moving the curve 2 units to the right transforms it to y = (x - 2)². Lastly, if shifted upwards by 2 units, the equation will be y = x² + 2.
