Question

numéro 3 (9 points)
soit les fonctions ( f(x) = 2 - x^2 ) et ( g(x) = \frac{x + 7}{3 - x} )
4 pts a) calculer ( (g circ f)(x) = g(f(x)) )
( g(f(x)) = \frac{(2 - x^2) + 7}{3 - (2 - x^2)} )
5 pts b) trouver ( g^{-1}(x) = )
( g^{-1}(x) = \frac{3 - x}{x + 7} )

Explanation:

Step 1: Calculate \((g \circ f)(x)\) (Part a)

To find the composition \(g(f(x))\), we substitute \(f(x) = 2 - x^2\) into \(g(x)=\frac{x + 7}{3 - x}\). So we replace every \(x\) in \(g(x)\) with \(2 - x^2\).

$$
LATEXBLOCK0
$$

Step 2: Find the inverse function \(g^{-1}(x)\) (Part b)

To find the inverse of \(g(x)=\frac{x + 7}{3 - x}\), we follow these steps:

1. Let \(y = g(x)=\frac{x + 7}{3 - x}\).
2. Swap \(x\) and \(y\): \(x=\frac{y + 7}{3 - y}\).
3. Solve for \(y\):

$$
LATEXBLOCK1
$$

Wait, there seems to be a mistake in the handwritten answer. Let's re - solve it:

Starting from \(x=\frac{y + 7}{3 - y}\)

Multiply both sides by \(3 - y\): \(x(3 - y)=y + 7\)

Expand: \(3x-xy=y + 7\)

Bring all terms with \(y\) to one side: \(3x - 7=y+xy\)

Factor \(y\) on the right - hand side: \(3x - 7=y(1 + x)\)

Then \(y=\frac{3x - 7}{x + 1}\) (the handwritten answer \(\frac{3 - x}{x + 7}\) is incorrect)

Answer:

a) \((g\circ f)(x)=\frac{9 - x^2}{1 + x^2}\)

b) \(g^{-1}(x)=\frac{3x - 7}{x + 1}\) (Note: The handwritten answer for part b is incorrect, the correct inverse is derived above)