Let f(x)=2x2−7x+5 and g(x)=x2+9.
Find f⋅g, and f/g
i don't understand how to do this.
Let f(x)=2x^2−7x+5 a IS A SQUARE
f "of" g = 2(x^2+9)^2-7(x^2+9) + 5
= 2x^2 + 18 - 7x^2 - 63 + 5
= - 5x^2 - 40
f / g = 2x^2-7x+5 / x^2 + 9
Factorise top and bottom
= (2x-5) (x-1) / (x+3)(x-3)
THis os a compositional function where:
f⋅g = f( (g)x) )
to solve it I substitute (x^2 + 9) for all values of x in the equation g(x) = 2x^2-7x+5
To find f⋅g (the product of f(x) and g(x)), you need to multiply the two functions together.
Step 1: Start by writing f(x) and g(x) as equations:
f(x) = 2x^2 - 7x + 5
g(x) = x^2 + 9
Step 2: Multiply each term of f(x) by each term of g(x) using the distributive property:
f⋅g = (2x^2 - 7x + 5)(x^2 + 9)
Step 3: Multiply each term from f(x) by each term from g(x):
f⋅g = (2x^2)(x^2) + (2x^2)(9) - (7x)(x^2) - (7x)(9) + (5)(x^2) + (5)(9)
Step 4: Simplify and combine like terms:
f⋅g = 2x^4 + 18x^2 - 7x^3 - 63x + 5x^2 + 45
So, f⋅g = 2x^4 - 7x^3 + 23x^2 - 63x + 45.
To find f/g (the quotient of f(x) and g(x)), you need to divide f(x) by g(x).
Step 1: Divide each term of f(x) by each term of g(x) using long division:
Note: Since the degree of the divisor (g(x)) is greater than the degree of the dividend (f(x)), the quotient will be 0.
..........................0
____________________________
x^2 + 9 | 2x^2 - 7x + 5
Step 2: Divide the first term of the dividend by the first term of the divisor:
0
Step 3: Multiply the divisor (x^2 + 9) by the quotient so far (0):
0(x^2 + 9) = 0
Step 4: Subtract the result obtained in step 3 from the dividend:
2x^2 - 7x + 5 - 0 = 2x^2 - 7x + 5
Step 5: Bring down the next term (if any):
Since the dividend has no more terms, the division is complete.
Therefore, f/g = 0.
In summary:
f⋅g = 2x^4 - 7x^3 + 23x^2 - 63x + 45
f/g = 0