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Math  /  Algebra

QuestionDetermine whether ff and gg are inverses by finding and simplifying f(g(x))f(g(x)) and g(f(x))g(f(x)). f(x)=3x−4g(x)=x3+4\begin{array}{l} f(x)=3 x-4 \\ g(x)=\frac{x}{3}+4 \end{array} f(g(x))=x+8f(g(x))=x+8 g(f(x))=3x−8g(f(x))=3 x-8
Are ff and gg inverses? Yes, they are inverses No, they are not inverses

Studdy Solution

STEP 1

1. Two functions f f and g g are inverses if and only if f(g(x))=x f(g(x)) = x and g(f(x))=x g(f(x)) = x for all x x in the domains of g g and f f respectively.
2. We need to verify both compositions f(g(x)) f(g(x)) and g(f(x)) g(f(x)) .

STEP 2

1. Find and simplify f(g(x)) f(g(x)) .
2. Find and simplify g(f(x)) g(f(x)) .
3. Determine if both compositions equal x x .

STEP 3

Substitute g(x) g(x) into f(x) f(x) to find f(g(x)) f(g(x)) :
g(x)=x3+4 g(x) = \frac{x}{3} + 4
f(g(x))=f(x3+4) f(g(x)) = f\left(\frac{x}{3} + 4\right)
=3(x3+4)−4 = 3\left(\frac{x}{3} + 4\right) - 4

STEP 4

Simplify f(g(x)) f(g(x)) :
=3⋅x3+3⋅4−4 = 3 \cdot \frac{x}{3} + 3 \cdot 4 - 4
=x+12−4 = x + 12 - 4
=x+8 = x + 8
Since f(g(x))=x+8≠x f(g(x)) = x + 8 \neq x , this does not satisfy the condition for inverse functions.

STEP 5

Substitute f(x) f(x) into g(x) g(x) to find g(f(x)) g(f(x)) :
f(x)=3x−4 f(x) = 3x - 4
g(f(x))=g(3x−4) g(f(x)) = g(3x - 4)
=3x−43+4 = \frac{3x - 4}{3} + 4

STEP 6

Simplify g(f(x)) g(f(x)) :
=3x3−43+4 = \frac{3x}{3} - \frac{4}{3} + 4
=x−43+123 = x - \frac{4}{3} + \frac{12}{3}
=x+83 = x + \frac{8}{3}
Since g(f(x))=x+83≠x g(f(x)) = x + \frac{8}{3} \neq x , this also does not satisfy the condition for inverse functions.

STEP 7

Since neither f(g(x))=x f(g(x)) = x nor g(f(x))=x g(f(x)) = x , f f and g g are not inverses of each other.
The answer is: No, they are not inverses.

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