Question

Determine which two functions are inverses of each other by function composition.(1 point)
Responses

f(x)=3x+79
and g(x)=3x−79
f left parenthesis x right parenthesis equals Start Fraction 3 x plus 7 over 9 End Fraction and g left parenthesis x right parenthesis equals Start Fraction 3 x minus 7 over 9 End Fraction

f(x)=12x−5
and g(x)=112x+5
f left parenthesis x right parenthesis equals 12 x minus 5 and g left parenthesis x right parenthesis equals Start Fraction 1 over 12 End Fraction x plus 5

f(x)=15x−13
and g(x)=x+1315
f left parenthesis x right parenthesis equals 15 x minus 13 and g left parenthesis x right parenthesis equals Start Fraction x plus 13 over 15 End Fraction

f(x)=11x−9
and g(x)=11x+9
f left parenthesis x right parenthesis equals 11 x minus 9 and g left parenthesis x right parenthesis equals 11 x plus 9

Answer 1

To determine which pairs of functions are inverses of each other through function composition, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \) for each pair.

1. **First pair:**
\( f(x) = 3x + 79 \)
\( g(x) = 3x - 79 \)

- \( f(g(x)) = f(3x - 79) = 3(3x - 79) + 79 = 9x - 237 + 79 = 9x - 158 \) (not equal to \( x \))
- \( g(f(x)) = g(3x + 79) = 3(3x + 79) - 79 = 9x + 237 - 79 = 9x + 158 \) (not equal to \( x \))

**Not inverses.**

2. **Second pair:**
\( f(x) = \frac{3x + 7}{9} \)
\( g(x) = \frac{3x - 7}{9} \)

- \( f(g(x)) = f\left(\frac{3x - 7}{9}\right) = \frac{3\left(\frac{3x - 7}{9}\right) + 7}{9} = \frac{\frac{9x - 21 + 63}{9}}{9} = \frac{9x + 42}{81} = x \) (this is incorrect)
- \( g(f(x)) = g\left(\frac{3x + 7}{9}\right) = \frac{3\left(\frac{3x + 7}{9}\right) - 7}{9} = \frac{9x + 21 - 63}{81} = x \) (this is incorrect)

**Not inverses.**

3. **Third pair:**
\( f(x) = 12x - 5 \)
\( g(x) = \frac{1}{12}x + 5 \)

- \( f(g(x)) = f\left(\frac{1}{12}x + 5\right) = 12\left(\frac{1}{12}x + 5\right) - 5 = x + 60 - 5 = x + 55 \) (not equal to \( x \))
- \( g(f(x)) = g(12x - 5) = \frac{1}{12}(12x - 5) + 5 = x - \frac{5}{12} + 5 = x + \frac{55}{12} \) (not equal to \( x \))

**Not inverses.**

4. **Fourth pair:**
\( f(x) = 11x - 9 \)
\( g(x) = 11x + 9 \)

- \( f(g(x)) = f(11x + 9) = 11(11x + 9) - 9 = 121x + 99 - 9 = 121x + 90 \) (not equal to \( x \))
- \( g(f(x)) = g(11x - 9) = 11(11x - 9) + 9 = 121x - 99 + 9 = 121x - 90 \) (not equal to \( x \))

**Not inverses.**

After checking all pairs, it seems that none of the pairs provided are inverses of each other. Would you like to check the calculations or explore different functions?