if $f(x) = -x^2 + 3x + 5$ and $g(x) = x^2 + 2x$, which graph shows the graph of $(f + g)(x)$?
if $f(x) = -x^2 + 3x + 5$ and $g(x) = x^2 + 2x$, which graph shows the graph of $(f + g)(x)$?
Answer
Explanation:
Step1: Find (f + g)(x)
To find ((f + g)(x)), we add the two functions (f(x)) and (g(x)) together. Given (f(x)=-x^{2}+3x + 5) and (g(x)=x^{2}+2x), we have: [ \begin{align*} (f + g)(x)&=f(x)+g(x)\ &=(-x^{2}+3x + 5)+(x^{2}+2x) \end{align*} ] Combine like - terms: The (-x^{2}) and (x^{2}) terms cancel out ((-x^{2}+x^{2}=0)), the (3x) and (2x) terms add up to (5x) ((3x + 2x = 5x)), and the constant term is (5). So ((f + g)(x)=5x + 5).
Step2: Analyze the function (y = 5x+5)
The function (y = 5x + 5) is a linear function in the form (y=mx + b), where the slope (m = 5) (a positive slope, so the line should be increasing from left to right) and the (y) - intercept (b = 5) (the line crosses the (y) - axis at ((0,5))).
Now we analyze the four graphs:
- The first graph: It is not a straight line (it looks like a curve), so it can't be the graph of a linear function.
- The second graph: It is a parabola (a quadratic function), not a linear function, so it is not the graph of (y = 5x+5).
- The third graph: It is a straight line with a positive slope and it crosses the (y) - axis at ((0,5)), which matches the characteristics of (y = 5x + 5).
- The fourth graph: It is a parabola opening downwards, not a linear function.