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

Responses
only resposes i gave shoul be use
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

1 answer

To determine which two functions are inverses of each other by function composition, we check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

  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 \( x \))

  2. Second pair:

    • \( f(x) = \frac{3x + 7}{9} \)
    • \( g(x) = \frac{3x - 7}{9} \)

    \( f(g(x)) = f\left(\frac{3x - 7}{9}\right) = \frac{3\left(\frac{3x - 7}{9}\right) + 7}{9} = \frac{\frac{9x - 21}{9} + 7}{9} = \frac{9x - 21 + 63}{81} = \frac{9x + 42}{81} \) (not equal to \( x \))

    \( g(f(x)) = g\left(\frac{3x + 7}{9}\right) = \frac{3\left(\frac{3x + 7}{9}\right) - 7}{9} = \frac{9x + 21 - 63}{81} = \frac{9x - 42}{81} \) (not equal to \( x \))

  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 \( x \))

  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 \( x \))

Upon checking all the pairs, it appears none of these functions are actually inverses of each other as shown with my calculations. Please verify the given functions to ensure they are entered correctly, or check whether they have undergone any modifications or input errors.