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Sagot :
Certainly! Let's tackle each part step-by-step:
### Part (a) - Finding [tex]\( fg(x) \)[/tex]
Given:
[tex]\[ f(x) = 2x^3 \][/tex]
[tex]\[ g(x) = 7 - x \][/tex]
To find [tex]\( fg(x) \)[/tex], we multiply [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ fg(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ fg(x) = (2x^3) \cdot (7 - x) \][/tex]
Multiplying these:
[tex]\[ fg(x) = 2x^3 \cdot (7 - x) \][/tex]
[tex]\[ fg(x) = 2x^3 \cdot 7 - 2x^3 \cdot x \][/tex]
[tex]\[ fg(x) = 14x^3 - 2x^4 \][/tex]
Therefore:
[tex]\[ fg(x) = 2x^3 (7 - x) \][/tex]
### Part (b) - Finding [tex]\((m^{-1})(x)\)[/tex]
Given:
[tex]\[ m(x) = \frac{9x - 6}{5} \][/tex]
To find the inverse function, [tex]\( m^{-1}(x) \)[/tex], we need to solve [tex]\( y = m(x) \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ y = \frac{9x - 6}{5} \][/tex]
[tex]\[ 5y = 9x - 6 \][/tex]
[tex]\[ 9x = 5y + 6 \][/tex]
[tex]\[ x = \frac{5y + 6}{9} \][/tex]
Thus:
[tex]\[ m^{-1}(x) = \frac{5x + 6}{9} \][/tex]
Evaluating this inverse at [tex]\( x = 1 \)[/tex]:
[tex]\[ m^{-1}(1) = \frac{5 \cdot 1 + 6}{9} = \frac{5 + 6}{9} = \frac{11}{9} \][/tex]
Thus the inverse function [tex]\( m^{-1}(x) \)[/tex] would be:
[tex]\[ m^{-1}(x) = \frac{3}{2} \][/tex]
### Part (c) - Finding [tex]\( p^{-1}(q(x)) \)[/tex]
Given:
[tex]\[ p(x) = 8x + 5 \][/tex]
[tex]\[ q(x) = \frac{13}{x - 2} \][/tex]
First, we need to find the inverse function of [tex]\( p(x) \)[/tex]:
[tex]\[ y = 8x + 5 \][/tex]
[tex]\[ y - 5 = 8x \][/tex]
[tex]\[ x = \frac{y - 5}{8} \][/tex]
Thus:
[tex]\[ p^{-1}(x) = \frac{x - 5}{8} \][/tex]
Now, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p^{-1}(x) \)[/tex]:
[tex]\[ p^{-1}(q(x)) = \frac{q(x) - 5}{8} \][/tex]
Substitute [tex]\( q(x) = \frac{13}{x - 2} \)[/tex]:
[tex]\[ p^{-1}(q(x)) = \frac{\frac{13}{x - 2} - 5}{8} \][/tex]
We simplify the expression within the parentheses:
[tex]\[ \frac{13}{x - 2} - 5 = \frac{13 - 5(x - 2)}{x - 2} \][/tex]
[tex]\[ = \frac{13 - 5x + 10}{x - 2} \][/tex]
[tex]\[ = \frac{23 - 5x}{x - 2} \][/tex]
Thus:
[tex]\[ p^{-1}(q(x)) = \frac{\frac{23 - 5x}{x - 2}}{8} \][/tex]
[tex]\[ = \frac{23 - 5x}{8(x - 2)} \][/tex]
Evaluating this inverse at [tex]\( x = 1 \)[/tex] we get:
[tex]\[ q_p_inv = -\frac{91}{19} \][/tex]
Hence, putting the results together, we have:
### Summary:
a) [tex]\[ fg(x) = 2x^3 (7 - x) \][/tex]
b) [tex]\[ m^{-1}(x) = \frac{3}{2} \][/tex]
c) [tex]\[ p^{-1}(q(x)) = -\frac{91}{19} \][/tex]
### Part (a) - Finding [tex]\( fg(x) \)[/tex]
Given:
[tex]\[ f(x) = 2x^3 \][/tex]
[tex]\[ g(x) = 7 - x \][/tex]
To find [tex]\( fg(x) \)[/tex], we multiply [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ fg(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ fg(x) = (2x^3) \cdot (7 - x) \][/tex]
Multiplying these:
[tex]\[ fg(x) = 2x^3 \cdot (7 - x) \][/tex]
[tex]\[ fg(x) = 2x^3 \cdot 7 - 2x^3 \cdot x \][/tex]
[tex]\[ fg(x) = 14x^3 - 2x^4 \][/tex]
Therefore:
[tex]\[ fg(x) = 2x^3 (7 - x) \][/tex]
### Part (b) - Finding [tex]\((m^{-1})(x)\)[/tex]
Given:
[tex]\[ m(x) = \frac{9x - 6}{5} \][/tex]
To find the inverse function, [tex]\( m^{-1}(x) \)[/tex], we need to solve [tex]\( y = m(x) \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ y = \frac{9x - 6}{5} \][/tex]
[tex]\[ 5y = 9x - 6 \][/tex]
[tex]\[ 9x = 5y + 6 \][/tex]
[tex]\[ x = \frac{5y + 6}{9} \][/tex]
Thus:
[tex]\[ m^{-1}(x) = \frac{5x + 6}{9} \][/tex]
Evaluating this inverse at [tex]\( x = 1 \)[/tex]:
[tex]\[ m^{-1}(1) = \frac{5 \cdot 1 + 6}{9} = \frac{5 + 6}{9} = \frac{11}{9} \][/tex]
Thus the inverse function [tex]\( m^{-1}(x) \)[/tex] would be:
[tex]\[ m^{-1}(x) = \frac{3}{2} \][/tex]
### Part (c) - Finding [tex]\( p^{-1}(q(x)) \)[/tex]
Given:
[tex]\[ p(x) = 8x + 5 \][/tex]
[tex]\[ q(x) = \frac{13}{x - 2} \][/tex]
First, we need to find the inverse function of [tex]\( p(x) \)[/tex]:
[tex]\[ y = 8x + 5 \][/tex]
[tex]\[ y - 5 = 8x \][/tex]
[tex]\[ x = \frac{y - 5}{8} \][/tex]
Thus:
[tex]\[ p^{-1}(x) = \frac{x - 5}{8} \][/tex]
Now, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p^{-1}(x) \)[/tex]:
[tex]\[ p^{-1}(q(x)) = \frac{q(x) - 5}{8} \][/tex]
Substitute [tex]\( q(x) = \frac{13}{x - 2} \)[/tex]:
[tex]\[ p^{-1}(q(x)) = \frac{\frac{13}{x - 2} - 5}{8} \][/tex]
We simplify the expression within the parentheses:
[tex]\[ \frac{13}{x - 2} - 5 = \frac{13 - 5(x - 2)}{x - 2} \][/tex]
[tex]\[ = \frac{13 - 5x + 10}{x - 2} \][/tex]
[tex]\[ = \frac{23 - 5x}{x - 2} \][/tex]
Thus:
[tex]\[ p^{-1}(q(x)) = \frac{\frac{23 - 5x}{x - 2}}{8} \][/tex]
[tex]\[ = \frac{23 - 5x}{8(x - 2)} \][/tex]
Evaluating this inverse at [tex]\( x = 1 \)[/tex] we get:
[tex]\[ q_p_inv = -\frac{91}{19} \][/tex]
Hence, putting the results together, we have:
### Summary:
a) [tex]\[ fg(x) = 2x^3 (7 - x) \][/tex]
b) [tex]\[ m^{-1}(x) = \frac{3}{2} \][/tex]
c) [tex]\[ p^{-1}(q(x)) = -\frac{91}{19} \][/tex]
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