The functions $f$ and $g$ are defined as follows:
[tex]f(x)=x^2+5 x+9[/tex] and [tex]g(x)=\frac{2}{x^2-3}[/tex]
Find [tex]f(x-2)[/tex] and [tex]g(-\frac{1}{x})[/tex]. Write your answers without parentheses and simplify them as much as possible.
[tex]f(x-2)=[/tex]
[tex]g(-\frac{1}{x})=[/tex]
Answer (2)
The simplified expression for f ( x − 2 ) is x 2 + x + 3 and for g ( − x 1 ) is 1 − 3 x 2 2 x 2 .
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To find f ( x − 2 ) , substitute x − 2 into f ( x ) : f ( x − 2 ) = ( x − 2 ) 2 + 5 ( x − 2 ) + 9 .
Expand and simplify f ( x − 2 ) : f ( x − 2 ) = x 2 − 4 x + 4 + 5 x − 10 + 9 = x 2 + x + 3 .
To find g ( − f r a c 1 x ) , substitute − f r a c 1 x into g ( x ) : g ( − f r a c 1 x ) = f r a c 2 ( − f r a c 1 x ) 2 − 3 = f r a c 2 f r a c 1 x 2 − 3 .
Simplify g ( − f r a c 1 x ) by multiplying the numerator and denominator by x 2 : g ( − f r a c 1 x ) = f r a c 2 x 2 1 − 3 x 2 .
The final answers are f ( x − 2 ) = x 2 + x + 3 and g ( − x 1 ) = 1 − 3 x 2 2 x 2 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 + 5 x + 9 and g ( x ) = f r a c 2 x 2 − 3 , and we need to find f ( x − 2 ) and g ( − f r a c 1 x ) . This involves substituting x − 2 into f ( x ) and − f r a c 1 x into g ( x ) , and then simplifying the resulting expressions.
Finding f(x-2) First, let's find f ( x − 2 ) . We substitute x − 2 for x in the expression for f ( x ) : f ( x − 2 ) = ( x − 2 ) 2 + 5 ( x − 2 ) + 9
Expanding the Expression Now, we expand the expression: f ( x − 2 ) = ( x 2 − 4 x + 4 ) + ( 5 x − 10 ) + 9
Simplifying f(x-2) Next, we combine like terms: f ( x − 2 ) = x 2 − 4 x + 5 x + 4 − 10 + 9
f ( x − 2 ) = x 2 + x + 3
Finding g(-1/x) Now, let's find g ( − f r a c 1 x ) . We substitute − f r a c 1 x for x in the expression for g ( x ) : g ( − f r a c 1 x ) = f r a c 2 ( − f r a c 1 x ) 2 − 3
Simplifying the Denominator Next, we simplify the denominator: g ( − f r a c 1 x ) = f r a c 2 f r a c 1 x 2 − 3
Simplifying g(-1/x) To get rid of the fraction in the denominator, we multiply both the numerator and the denominator by x 2 : g ( − f r a c 1 x ) = f r a c 2 x 2 1 − 3 x 2
Final Answer Therefore, f ( x − 2 ) = x 2 + x + 3 and g ( − f r a c 1 x ) = f r a c 2 x 2 1 − 3 x 2 .
Examples
Understanding function transformations is crucial in many real-world applications. For instance, in physics, if you have a function that describes the position of an object over time, f ( t ) , then f ( t − 2 ) would describe the position of the object as if you started observing it 2 seconds later. Similarly, if you have a function describing the intensity of light, g ( x ) , then g ( − f r a c 1 x ) might represent how the intensity changes when viewed through a lens that inverts and scales the image. These transformations help us analyze and predict behavior in various scenarios.
