h(x)=5/x^2+6
I know the answer is: f(x)=x+6, g(x)=5/x^2, but I do not know how to get it. Can someone help me understand it?
First break it down
x=x+6 and x=5/x^2
x=-1 x=2.5
You did not state the question.
Is it
h of f of g of x = 5/x^2 + 6 ?
There is surely more than one answer you can find by trial and error. For example another might be:
try g(x) = 1/x^2
then f(x) = 5 x
then f (g) = 5 (1/x^2) = 5/x^2 which is what I want except I have to add 6
so say h(x) = x + 6
then h (f) = 5/x^2 + 6
so the sequence
g(x) = 1/x^2
f(x) = 5 x
h(x) =x + 6
also yields
h(x) = 5/x^2 + 6
Consider the function h as defined. Find functions f and g so that
(f o g)(x)=h(x)
h(x)=5/x^2+6
oh
then I would modify that answer:
There is surely more than one answer you can find by trial and error. For example another might be:
try g(x) = 1/x^2
then f(x) = 5 x + 6
then f (g) = 5 (1/x^2) +6 = 5/x^2 + 6
then h (f) = 5/x^2 + 6
so the sequence
g(x) = 1/x^2
f(x) = 5 x + 6
also yields
h(x) = 5/x^2 + 6
To find the answer, we need to simplify the expression h(x) = 5/x^2 + 6.
Step 1: Start by factoring out the common factor in the numerator, which is 5. This gives us h(x) = 5(1/x^2) + 6.
Step 2: Simplify the expression 1/x^2. To do this, we can rewrite it as (1/x) * (1/x), which is equivalent to x^(-1) * x^(-1) using the property x^(-n) = 1/x^n. Therefore, 1/x^2 can be rewritten as (x^(-1))^2.
Step 3: Substitute the rewritten expression into the original equation. This gives us h(x) = 5(x^(-1))^2 + 6.
Step 4: Simplify further by squaring the expression x^(-1). When we square a term with a negative exponent, the negative exponent becomes positive. Therefore, (x^(-1))^2 is equivalent to x^(-2).
Step 5: Substitute the simplified expression back into the equation. Now we have h(x) = 5x^(-2) + 6.
Step 6: To simplify the expression further, we can rewrite x^(-2) as 1/x^2, using the property x^(-n) = 1/x^n. Now we have h(x) = 5(1/x^2) + 6.
Step 7: Combine like terms by adding the fractions with the same denominator. The fractions 5(1/x^2) and 6 have a common denominator of x^2. This gives us h(x) = (5 + 6x^2)/x^2.
Step 8: The expression (5 + 6x^2)/x^2 can be further simplified by factoring out x^2 from the numerator. This leads to h(x) = (5/x^2) + (6x^2/x^2).
Step 9: Simplify the fractions separately. The fraction 5/x^2 simplifies to 5 * (1/x^2), which is equivalent to 5/x^2. The fraction 6x^2/x^2 simplifies to 6 * (x^2/x^2), which simplifies to 6 * 1, which is just 6.
Step 10: Combine the simplified fractions to obtain the final answer: h(x) = 5/x^2 + 6 = f(x) + g(x) = (5/x^2) + 6.
Therefore, the functions f(x) and g(x) are f(x) = 5/x^2 and g(x) = 6.