Math Problem Statement
f(x)=x^2-9 for x ∈ R, x<0 g(x)=10-x/2 for x ∈ R, x>6 findthe range of fg(x)
Solution
To find the range of , we will analyze the composition of and under the given constraints.
Step 1: Determine the domain of
Given for , the output of is:
So .
Step 2: Apply to
We now evaluate . Substituting into , we get:
Simplify :
Step 3: Range of
The domain of implies , but is defined only for . Thus, for , we need to fall in the part of corresponding to .
- From , solve for .
Finalizing the Range of :
We are tasked with finding the range of , where , and is defined only for .
Step 1: Analyze for
The function is a parabola that opens upwards, symmetric about , and restricted to . For , decreases as approaches from the left. Thus:
Step 2: Analyze for
The function decreases linearly for . The domain implies , meaning:
Step 3: Composition
For , substitute into . Since is defined for , we must check where maps back into the valid input domain of .
a. Solve for :
Let . For , must also be .
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Math Problem Analysis
Mathematical Concepts
Function Composition
Quadratic Functions
Piecewise Functions
Formulas
f(x) = x^2 - 9
g(x) = 10 - x/2
f(g(x)) = (10 - x/2)^2 - 9
Theorems
Properties of Quadratic Functions
Domain and Range of Functions
Suitable Grade Level
Grades 11-12
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