Find f(f(f(f(..f(10)..., where the above expression contains 97 fs and f(x) = 5/x

f(f(f(f(...f(10)..., where the expression is 97 f(10) is equal to 1/2.

Step-by-step explanation:

For this problem, let's simply solve a few of the entry cases and find a pattern that we can scale.  Given that f(x) = 5/x

f(x) = 5/x

f(f(x)) = 5/(5/x) = 5 * (x/5) = x

f(f(f(x))) = 5/(5/(5/x)) = 5/(5* x/5) = 5/(x) = 5/x

f(f(f(f(x = 5/(5/(5/(5/x))) = 5/(5/(5* x/5) = 5/(5/x) = 5*x/5 = x

After solving a few of the initial cases, we have found a pattern.  Notice, that when the number of functions of f are odd, the resultant is 5/x and when the number of functions of f are even, the resultant is x.  Using this pattern we can say that our functions of f(10), if the number is 97, will follow the odd pattern since 97 is an odd number.

So we can say the following:

f(f(f(f(...f(x)... = 5/x

f(f(f(f(...f(10)... = 5/(10) = 5/10 = 5 * 1 / 5 * 2 = 1/2

Hence, we have found that if we evaluate 10 inside of 97 of the function of x as defined, we will have the value 1/2.

Cheers.

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)

if gloria has "carry over" minutes, (that is, any minutes paid for but not used in a month can be applied to her payment for minutes in the next month,) her total usage for the three months is: total minutes = 320 + 243 + 489 = 1052 minutes number of 200 minute blocks = 1052/200 = 5 r52 (that is 5 with a remainder of 52) rounding up the number of 200 minute blocks gives her usage for the three months as 6 blocks. the cost for six blocks is 6 x $25 = $150 she also has 200 - 52 or 148 carry over minutes if gloria pays each month and unused minutes are lost, applying the rounding up procedure to each month gives: first month usage of 320 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 second month usage of 243 minutes rounds up to 2 blocks of 200 minutes for a payment of 2 x $25 = $50 third month usage of 489 minutes rounds up to 3 blocks of 200 minutes for a payment of 3 x $25 = $75 total payments for the three months are $50 + $50 + $75 = $175 get the plan with carry over minutes!