Let f(x) = (3x2 + x - 5) / (x2 + 4). Find f(3), f(2), f(1), f(a+1), f(sqrt(a)), f(a2), f(a) + f(1).

Mathematics
Let f(x) = (3x2 + x - 5) / (x2 + 4). Find f(3), f(2), f(1), f(a+1), f(sqrt(a)), f(a2), f(a) + f(1).

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Answer

a2+b2+2ab+2a+2b−4a^2 + b^2 + 2ab + 2a + 2b - 4

Step 1: Understand the given function. We are given the function h(x)=x2+2x−4h(x) = x^2 + 2x - 4.

Step 2: Solve part a). a) Find h(b+a)h(b+a). Substitute x=(b+a)x = (b+a) into the function: h(b+a)=(b+a)2+2(b+a)−4h(b+a) = (b+a)^2 + 2(b+a) - 4 Expand the terms: h(b+a)=(b2+2ab+a2)+(2b+2a)−4h(b+a) = (b^2 + 2ab + a^2) + (2b + 2a) - 4 h(b+a)=a2+b2+2ab+2a+2b−4h(b+a) = a^2 + b^2 + 2ab + 2a + 2b - 4 The simplified expression is a2+b2+2ab+2a+2b−4\boxed{a^2 + b^2 + 2ab + 2a + 2b - 4}.

Step 3: Solve part b). b) Find h(a−b)h(a-b). Substitute x=(a−b)x = (a-b) into the function: h(a−b)=(a−b)2+2(a−b)−4h(a-b) = (a-b)^2 + 2(a-b) - 4 Expand the terms: h(a−b)=(a2−2ab+b2)+(2a−2b)−4h(a-b) = (a^2 - 2ab + b^2) + (2a - 2b) - 4 h(a−b)=a2+b2−2ab+2a−2b−4h(a-b) = a^2 + b^2 - 2ab + 2a - 2b - 4 The simplified expression is a2+b2−2ab+2a−2b−4\boxed{a^2 + b^2 - 2ab + 2a - 2b - 4}.

Step 4: Solve part c). c) Find h(1+b)−h(1)b\frac{h(1+b)-h(1)}{b}. First, find h(1+b)h(1+b): h(1+b)=(1+b)2+2(1+b)−4h(1+b) = (1+b)^2 + 2(1+b) - 4 h(1+b)=(1+2b+b2)+(2+2b)−4h(1+b) = (1 + 2b + b^2) + (2 + 2b) - 4 h(1+b)=b2+4b+3−4h(1+b) = b^2 + 4b + 3 - 4 h(1+b)=b2+4b−1h(1+b) = b^2 + 4b - 1 Next, find h(1)h(1): h(1)=(1)2+2(1)−4h(1) = (1)^2 + 2(1) - 4 h(1)=1+2−4h(1) = 1 + 2 - 4 h(1)=−1h(1) = -1 Now, substitute these into the expression: h(1+b)−h(1)b=(b2+4b−1)−(−1)b\frac{h(1+b)-h(1)}{b} = \frac{(b^2 + 4b - 1) - (-1)}{b} h(1+b)−h(1)b=b2+4b−1+1b\frac{h(1+b)-h(1)}{b} = \frac{b^2 + 4b - 1 + 1}{b} h(1+b)−h(1)b=b2+4bb\frac{h(1+b)-h(1)}{b} = \frac{b^2 + 4b}{b} Factor out bb from the numerator: h(1+b)−h(1)b=b(b+4)b\frac{h(1+b)-h(1)}{b} = \frac{b(b+4)}{b} Cancel bb (assuming b≠0b \neq 0): h(1+b)−h(1)b=b+4\frac{h(1+b)-h(1)}{b} = b+4 The simplified expression is b+4\boxed{b+4}.

Step 5: Solve part d). d) Find h(a+b)−h(a)b\frac{h(a+b)-h(a)}{b}. First, find h(a+b)h(a+b): h(a+b)=(a+b)2+2(a+b)−4h(a+b) = (a+b)^2 + 2(a+b) - 4 h(a+b)=a2+2ab+b2+2a+2b−4h(a+b) = a^2 + 2ab + b^2 + 2a + 2b - 4 Next, find h(a)h(a): h(a)=a2+2a−4h(a) = a^2 + 2a - 4 Now, substitute these into the expression: h(a+b)−h(a)b=(a2+2ab+b2+2a+2b−4)−(a2+2a−4)b\frac{h(a+b)-h(a)}{b} = \frac{(a^2 + 2ab + b^2 + 2a + 2b - 4) - (a^2 + 2a - 4)}{b} Distribute the negative sign in the numerator: h(a+b)−h(a)b=a2+2ab+b2+2a+2b−4−a2−2a+4b\frac{h(a+b)-h(a)}{b} = \frac{a^2 + 2ab + b^2 + 2a + 2b - 4 - a^2 - 2a + 4}{b} Combine like terms in the numerator: h(a+b)−h(a)b=(a2−a2)+(2a−2a)+(−4+4)+2ab+b2+2bb\frac{h(a+b)-h(a)}{b} = \frac{(a^2 - a^2) + (2a - 2a) + (-4 + 4) + 2ab + b^2 + 2b}{b} h(a+b)−h(a)b=2ab+b2+2bb\frac{h(a+b)-h(a)}{b} = \frac{2ab + b^2 + 2b}{b} Factor out bb from the numerator: h(a+b)−h(a)b=b(2a+b+2)b\frac{h(a+b)-h(a)}{b} = \frac{b(2a + b + 2)}{b} Cancel bb (assuming b≠0b \neq 0): h(a+b)−h(a)b=2a+b+2\frac{h(a+b)-h(a)}{b} = 2a + b + 2 The simplified expression is 2a+b+2\boxed{2a + b + 2}.

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Understand the given function. We are given the function h(x) = x^2 + 2x - 4.

Let f(x) = (3x2 + x - 5) / (x2 + 4). Find f(3), f(2), f(1), f(a+1), f(sqrt(a)), f(a2), f(a) + f(1).
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Understand the given function. We are given the function h(x) = x^2 + 2x - 4. Step 2: Solve part a). a) Find h(b+a). Substitute x = (b+a) into the function: h(b+a) = (b+a)^2 + 2(b+a) - 4 Expand the terms: h(b+a) = (b^2 + 2ab + a^2) + (2b + 2a) - 4 h(b+a) = a^2 + b^2 + 2ab + 2a + 2b - 4 The simplified expression is a^2 + b^2 + 2ab + 2a + 2b - 4. Step 3: Solve part b). b) Find h(a-b). Substitute x = (a-b) into the function: h(a-b) = (a-b)^2 + 2(a-b) - 4 Expand the terms: h(a-b) = (a^2 - 2ab + b^2) + (2a - 2b) - 4 h(a-b) = a^2 + b^2 - 2ab + 2a - 2b - 4 The simplified expression is a^2 + b^2 - 2ab + 2a - 2b - 4. Step 4: Solve part c). c) Find (h(1+b)-h(1))/(b). First, find h(1+b): h(1+b) = (1+b)^2 + 2(1+b) - 4 h(1+b) = (1 + 2b + b^2) + (2 + 2b) - 4 h(1+b) = b^2 + 4b + 3 - 4 h(1+b) = b^2 + 4b - 1 Next, find h(1): h(1) = (1)^2 + 2(1) - 4 h(1) = 1 + 2 - 4 h(1) = -1 Now, substitute these into the expression: (h(1+b)-h(1))/(b) = ((b^2 + 4b - 1) - (-1))/(b) (h(1+b)-h(1))/(b) = (b^2 + 4b - 1 + 1)/(b) (h(1+b)-h(1))/(b) = (b^2 + 4b)/(b) Factor out b from the numerator: (h(1+b)-h(1))/(b) = (b(b+4))/(b) Cancel b (assuming b ≠ 0): (h(1+b)-h(1))/(b) = b+4 The simplified expression is b+4. Step 5: Solve part d). d) Find (h(a+b)-h(a))/(b). First, find h(a+b): h(a+b) = (a+b)^2 + 2(a+b) - 4 h(a+b) = a^2 + 2ab + b^2 + 2a + 2b - 4 Next, find h(a): h(a) = a^2 + 2a - 4 Now, substitute these into the expression: (h(a+b)-h(a))/(b) = ((a^2 + 2ab + b^2 + 2a + 2b - 4) - (a^2 + 2a - 4))/(b) Distribute the negative sign in the numerator: (h(a+b)-h(a))/(b) = (a^2 + 2ab + b^2 + 2a + 2b - 4 - a^2 - 2a + 4)/(b) Combine like terms in the numerator: (h(a+b)-h(a))/(b) = ((a^2 - a^2) + (2a - 2a) + (-4 + 4) + 2ab + b^2 + 2b)/(b) (h(a+b)-h(a))/(b) = (2ab + b^2 + 2b)/(b) Factor out b from the numerator: (h(a+b)-h(a))/(b) = (b(2a + b + 2))/(b) Cancel b (assuming b ≠ 0): (h(a+b)-h(a))/(b) = 2a + b + 2 The simplified expression is 2a + b + 2.