Mathematics Exam > Mathematics Questions > If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, the...
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If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, then function g(x) is
- a)continuous on R but not differentiable
- b)Neither continuous nor differentiable on R
- c)Both continuous and differentiable on R
- d)Continuos only is some bounded region of R.
Correct answer is option 'C'. Can you explain this answer?
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If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, then function g(x) isa)conti...
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If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, then function g(x) isa)conti...
Explanation:
To determine the properties of function g(x), we need to analyze the given information step by step.
Given:
f(x) = 3x - 2
(gof)-1(x) = x - 2
Step 1: Composition of Functions
Let's first find the composition of functions gof(x):
gof(x) = g(f(x))
Substituting the expression for f(x):
gof(x) = g(3x - 2)
Step 2: Inverse of Composition
Now, we are given that the inverse of gof(x) is (gof)-1(x) = x - 2.
From this, we can conclude that:
g(3x - 2) = x - 2
Step 3: Solving for g(x)
To determine the function g(x), we need to solve the equation obtained in the previous step for g(x).
g(3x - 2) = x - 2
Let's substitute u = 3x - 2 for simplicity:
g(u) = (1/3)u + 2
Therefore, the function g(x) is:
g(x) = (1/3)(3x - 2) + 2
g(x) = x + (2/3)
Step 4: Analyzing the Properties of g(x)
a) Continuity:
To check the continuity of g(x), we need to observe that the function is a sum of continuous functions (polynomials) and does not contain any points of discontinuity (such as division by zero or square roots of negative numbers). Hence, g(x) is continuous on the entire real line, R.
b) Differentiability:
Since g(x) is a linear function (a polynomial of degree 1), it is differentiable everywhere. The derivative of g(x) is simply 1, indicating a constant slope or rate of change.
Conclusion:
Based on the analysis, we can conclude that function g(x) is both continuous and differentiable on the entire real line, R. Therefore, the correct answer is option C) Both continuous and differentiable on R.
To determine the properties of function g(x), we need to analyze the given information step by step.
Given:
f(x) = 3x - 2
(gof)-1(x) = x - 2
Step 1: Composition of Functions
Let's first find the composition of functions gof(x):
gof(x) = g(f(x))
Substituting the expression for f(x):
gof(x) = g(3x - 2)
Step 2: Inverse of Composition
Now, we are given that the inverse of gof(x) is (gof)-1(x) = x - 2.
From this, we can conclude that:
g(3x - 2) = x - 2
Step 3: Solving for g(x)
To determine the function g(x), we need to solve the equation obtained in the previous step for g(x).
g(3x - 2) = x - 2
Let's substitute u = 3x - 2 for simplicity:
g(u) = (1/3)u + 2
Therefore, the function g(x) is:
g(x) = (1/3)(3x - 2) + 2
g(x) = x + (2/3)
Step 4: Analyzing the Properties of g(x)
a) Continuity:
To check the continuity of g(x), we need to observe that the function is a sum of continuous functions (polynomials) and does not contain any points of discontinuity (such as division by zero or square roots of negative numbers). Hence, g(x) is continuous on the entire real line, R.
b) Differentiability:
Since g(x) is a linear function (a polynomial of degree 1), it is differentiable everywhere. The derivative of g(x) is simply 1, indicating a constant slope or rate of change.
Conclusion:
Based on the analysis, we can conclude that function g(x) is both continuous and differentiable on the entire real line, R. Therefore, the correct answer is option C) Both continuous and differentiable on R.
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If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, then function g(x) isa)continuous on R but not differentiableb)Neither continuous nor differentiable on Rc)Both continuous and differentiable on Rd)Continuos only is some bounded region of R.Correct answer is option 'C'. Can you explain this answer?
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