Real functions and economic models

Find the natural domain of $f(x)=\frac{x}{x^2-5x+6}$.

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Find the natural domain of $f(x)=\sqrt{9-x^2}$.

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Find the natural domain of $f(x)=\ln \frac{x+1}{3-x}$.

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Find the inverse of $f(x)=\frac{2x-3}{5}$ and verify that $f^{-1}(f(1))=1$.

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Find the inverse of $f(x)=\ln (x-2)+1$, defined for $x>2$, with its domain.

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Let $f(x)=x^2+1$ and $g(x)=\sqrt{x-2}$. Write $g\circ f$ and $f\circ g$ and find their domains.

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Find the linear function whose graph passes through $(-1,4)$ and $(3,12)$, with its slope and its intercept.

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A firm spends $2600$ euros to produce $80$ pieces and $3400$ euros to produce $120$. Assuming the cost is linear, find the fixed cost and the variable unit cost.

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Write the equation of the line that passes through the point $(2,-1)$ and is parallel to the line $y=-3x+4$.

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The demand and supply of a good are $q_d(p)=200-5p$ and $q_s(p)=3p-40$. Find the equilibrium price and quantity.

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A product has fixed costs of $12000$ euros, a variable cost of $35$ euros per unit and a selling price of $50$ euros. Find the break-even point.

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Find the vertex of the parabola $f(x)=2x^2-8x+3$ and the minimum value of $f$.

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A firm has cost $C(q)=500+20q$ and inverse demand $p=100-q$. Find the quantity that maximizes profit, the maximum profit and the break-even points.

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A capital of $3000$ euros is invested at $4\%$ per year with compound interest. Compute its value after $5$ years and the time needed for it to double.

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