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Is f(x) f(x) the inverse function of g(x) g(x) ?\newlinef(x)=x f(x) = x \newlineg(x)=−4x g(x) = -4x \newlineChoices:\newline[yes] \text{[yes]} \newline[no] \text{[no]}

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Q. Is f(x) f(x) the inverse function of g(x) g(x) ?\newlinef(x)=x f(x) = x \newlineg(x)=−4x g(x) = -4x \newlineChoices:\newline[yes] \text{[yes]} \newline[no] \text{[no]}
  1. Calculate (f∘g)(x)(f\circ g)(x): To check if f(x)f(x) is the inverse of g(x)g(x), calculate (f∘g)(x)(f\circ g)(x) and see if it equals xx.\newline(f∘g)(x)=f(g(x))=f(−4x)=−4x(f\circ g)(x) = f(g(x)) = f(-4x) = -4x
  2. Calculate (g∘f)(x)(g\circ f)(x): Now, calculate (g∘f)(x)(g\circ f)(x) and see if it equals xx.(g∘f)(x)=g(f(x))=g(x)=−4x(g\circ f)(x) = g(f(x)) = g(x) = -4x
  3. Check for Inverses: Check if (f∘g)(x)=x(f\circ g)(x) = x and (g∘f)(x)=x(g\circ f)(x) = x for f(x)f(x) and g(x)g(x) to be inverses.\newline(f∘g)(x)=−4x≠x(f\circ g)(x) = -4x \neq x and (g∘f)(x)=−4x≠x(g\circ f)(x) = -4x \neq x\newlineSo, f(x)f(x) is not the inverse of g(x)g(x).

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