Find left parenthesis f dot g right parenthesis left parenthesis x right parenthesis when f left parenthesis x right parenthesis equals x squared plus 8 x plus 15 and g left parenthesis x right parenthesis equals start fraction 5 over x squared minus 9 end fraction.
To find f(g(x)), we need to substitute g(x) into f(x).
First, let's find g(x).
g(x) = (5 / (x^2 - 9))
Now, substitute g(x) into f(x):
f(g(x)) = f(5 / (x^2 - 9))
f(x) = x^2 + 8x + 15
f(g(x)) = (5 / (x^2 - 9))^2 + 8(5 / (x^2 - 9)) + 15
Simplifying further:
f(g(x)) = (25 / (x^2 - 9)^2) + (40 / (x^2 - 9)) + 15
Therefore, the expression f(g(x)) = (25 / (x^2 - 9)^2) + (40 / (x^2 - 9)) + 15
options
a) left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 x plus 25 over x minus 3 end fractionImage with alt text: left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 x plus 25 over x minus 3 end fraction
b) left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 over x superscript 4 baseline plus 16 x cubed plus 94 x squared plus 240 x plus 216 end fraction
Image with alt text: left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 over x superscript 4 baseline plus 16 x cubed plus 94 x squared plus 240 x plus 216 end fraction
c) left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 x plus 25 over x plus 3 end fraction
Image with alt text: left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 5 x plus 25 over x plus 3 end fraction
d) left parenthesis f dot g right parenthesis left parenthesis x right parenthesis equals start fraction 25 over x superscript 4 baseline minus 18 x squared plus 81 end fraction plus start fraction 40 over x squared minus 9 end fraction plus 15
The correct option is:
d) (f ∘ g)(x) = (25 / (x^4 - 18x^2 + 81)) + (40 / (x^2 - 9)) + 15
Find the inverse of the function f left parenthesis x right parenthesis equals start fraction 7 x minus 4 over x plus 3 end fraction.
To find the inverse of the function f(x) = (7x - 4) / (x + 3), we can follow these steps:
Step 1: Replace f(x) with y.
y = (7x - 4) / (x + 3)
Step 2: Swap x and y.
x = (7y - 4) / (y + 3)
Step 3: Solve for y.
Multiply both sides by (y + 3) to get rid of the denominator.
x(y + 3) = 7y - 4
xy + 3x = 7y - 4
3x + 4 = 7y - xy
Now, isolate y.
3x + 4 = (7 - x)y
y = (3x + 4) / (7 - x)
The inverse of the function f(x) = (7x - 4) / (x + 3) is given by:
f^(-1)(x) = (3x + 4) / (7 - x)
To find f(g(x)), we need to substitute the expression for g(x) into f(x):
f(g(x)) = f(5/(x^2 - 9))
Now, let's substitute g(x) in place of x in the expression for f(x):
f(g(x)) = (5/(x^2 - 9))^2 + 8 * (5/(x^2 - 9)) + 15
Next, let's simplify the expression further:
f(g(x)) = 25/(x^4 - 18x^2 + 81) + 40/(x^2 - 9) + 15
So, the final expression for f(g(x)) is:
f(g(x)) = 25/(x^4 - 18x^2 + 81) + 40/(x^2 - 9) + 15
To find f(g(x)), we need to substitute g(x) into f(x).
Given that f(x) = x^2 + 8x + 15 and g(x) = 5/(x^2 - 9), we substitute g(x) into f(x):
f(g(x)) = f(5/(x^2 - 9))
Next, we need to substitute the expression for g(x) into f(x):
f(g(x)) = (5/(x^2 - 9))^2 + 8(5/(x^2 - 9)) + 15
Now, we simplify the expression:
f(g(x)) = 25/(x^2 - 9)^2 + 40/(x^2 - 9) + 15
Therefore, f(g(x)) = 25/(x^2 - 9)^2 + 40/(x^2 - 9) + 15.