Find f(n) - g(n) given f(n) = x^2 + 1 and g(n) = 2x - 5.
Understand the Problem
The question is asking to find the difference between two functions, f(n) and g(n), which involves substituting the provided expressions into the equation. We will subtract g(n) from f(n) and simplify the result.
Answer
The final answer is \( x^2 - 2x + 6 \).
Answer for screen readers
The result of ( f(n) - g(n) ) is ( x^2 - 2x + 6 ).
Steps to Solve
- Identify the expressions for f(n) and g(n)
Here we have:
- ( f(n) = x^2 + 1 )
- ( g(n) = 2x - 5 )
- Set up the subtraction of the functions
We want to find ( f(n) - g(n) ), so we write: $$ f(n) - g(n) = (x^2 + 1) - (2x - 5) $$
- Distribute the negative sign
Distributing the negative sign changes ( g(n) ): $$ f(n) - g(n) = x^2 + 1 - 2x + 5 $$
- Combine like terms
Now we combine the constant terms and the terms involving ( x ): $$ f(n) - g(n) = x^2 - 2x + (1 + 5) = x^2 - 2x + 6 $$
The result of ( f(n) - g(n) ) is ( x^2 - 2x + 6 ).
More Information
This result represents the difference between the two functions. The resulting polynomial has a quadratic term, a linear term, and a constant, which indicates that the graph of ( f(n) - g(n) ) is a parabola.
Tips
- Forgetting to distribute the negative sign correctly can lead to sign errors in the terms.
- Not combining like terms accurately towards the end can result in a wrong answer.
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