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If a function f(x) is defined as h(x+1), where h(x) is a real function. g(x) is defined as f(x+1). If g(x+1) = x2 + 6x + 10x
Find the value of h(5).
Find the value of h(5).
- a)26
- b)25
- c)50
- d)51
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If a function f(x) is defined as h(x+1), where h(x) is a real function...
g(x+1) = x2 + 6x + 10
g(x+1) = x2 + 6x +9 +1 = (x + 3)2 + 1
g(x+1) = (x + 1 + 2)2 + 1
g(x) = (x + 2)2 + 1
f(x+1) = g(x) = (x + 2)2 +1
f(x+1) = (x + 1 + 1)2 +1
f(x) = (x + 1)2 + 1
h(x+1) = f(x) = (x + 1)2 + 1
h(x) = x2 +1
Hence, h(5) = 26
g(x+1) = x2 + 6x +9 +1 = (x + 3)2 + 1
g(x+1) = (x + 1 + 2)2 + 1
g(x) = (x + 2)2 + 1
f(x+1) = g(x) = (x + 2)2 +1
f(x+1) = (x + 1 + 1)2 +1
f(x) = (x + 1)2 + 1
h(x+1) = f(x) = (x + 1)2 + 1
h(x) = x2 +1
Hence, h(5) = 26
Community Answer
If a function f(x) is defined as h(x+1), where h(x) is a real function...
To find the value of h(5), we need to understand the relationship between the functions f(x) and g(x), and then use the given information about g(x) to solve for h(5).
Relationship between f(x) and g(x):
f(x) is defined as h(x+1), which means that f(x) takes the input x and adds 1 to it before applying the function h(x). In other words, f(x) = h(x+1).
g(x) is defined as f(x+1), which means that g(x) takes the input x and adds 1 to it before applying the function f(x). In other words, g(x) = f(x+1) = h(x+1+1) = h(x+2).
Given information about g(x):
g(x+1) = x^2 - 6x + 10x
We can substitute x+1 for x in the expression for g(x) to find g(x+1):
g(x+1+1) = (x+1)^2 - 6(x+1) + 10(x+1)
g(x+2) = x^2 + 2x + 1 - 6x - 6 + 10x + 10
g(x+2) = x^2 + 6x + 5
Now we can equate the expression for g(x+2) with the given expression for g(x+1):
x^2 + 6x + 5 = x^2 - 6x + 10x
Simplifying the equation:
6x + 5 = 4x
2x = -5
x = -5/2
Since g(x) = f(x+1) = h(x+2), we can substitute x = -5/2 into the expression for g(x):
g(-5/2) = h(-5/2+2) = h(-1/2)
Now we need to find h(5), so we need to find the value of h(x) when x = -1/2. However, we are given that h(x) is a real function, which means it does not have any complex or imaginary values. Therefore, h(x) must be undefined for x = -1/2.
Since h(x) is undefined for x = -1/2, it means that h(5) is also undefined. Therefore, none of the given options (a, b, c, d) are correct.
Relationship between f(x) and g(x):
f(x) is defined as h(x+1), which means that f(x) takes the input x and adds 1 to it before applying the function h(x). In other words, f(x) = h(x+1).
g(x) is defined as f(x+1), which means that g(x) takes the input x and adds 1 to it before applying the function f(x). In other words, g(x) = f(x+1) = h(x+1+1) = h(x+2).
Given information about g(x):
g(x+1) = x^2 - 6x + 10x
We can substitute x+1 for x in the expression for g(x) to find g(x+1):
g(x+1+1) = (x+1)^2 - 6(x+1) + 10(x+1)
g(x+2) = x^2 + 2x + 1 - 6x - 6 + 10x + 10
g(x+2) = x^2 + 6x + 5
Now we can equate the expression for g(x+2) with the given expression for g(x+1):
x^2 + 6x + 5 = x^2 - 6x + 10x
Simplifying the equation:
6x + 5 = 4x
2x = -5
x = -5/2
Since g(x) = f(x+1) = h(x+2), we can substitute x = -5/2 into the expression for g(x):
g(-5/2) = h(-5/2+2) = h(-1/2)
Now we need to find h(5), so we need to find the value of h(x) when x = -1/2. However, we are given that h(x) is a real function, which means it does not have any complex or imaginary values. Therefore, h(x) must be undefined for x = -1/2.
Since h(x) is undefined for x = -1/2, it means that h(5) is also undefined. Therefore, none of the given options (a, b, c, d) are correct.
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