f⁻¹(x) = ¹/₂ x - ³/₂
Further explanation
Option 1
\(\boxed{ \ f(x) = 2x + 3} \rightarrow f^{-1}(x)? \ }\)
Let \(\boxed{y = 2x + 3}\)
Rewrite and set the function with subject x.
Both sides are subtracted by 3.
y - 3 = 2x + 3 - 3
y - 3 = 2x
2x = y - 3
Both sides are divided by 2.
\(\boxed{x = \frac{1}{2}y - \frac{3}{2}}\)
Let's replace \(\boxed{x \ with \ f^{-1}(x)}\) and \(\boxed{y \ with \ x}\).
Thus, the inverse of f(x) = 2x + 3 is \(\boxed{\boxed{ \ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \ }}\)
Option 2
\(\boxed{ \ f(x) = 2x + 3} \rightarrow x = 2f^{-1}(x) + 3 \ }\)
\(\boxed{ \ x - 3 = 2f^{-1}(x) \ }\)
\(\boxed{ \ 2f^{-1}(x) = x - 3\ }\)
\(\boxed{\boxed{ \ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \ }}\)
Note:
- A function has an inverse if, and only if, it is one-to-one and onto.
- A function f is one-to-one if, for each x₁ ≠ x₂, in its domain, f(x₁) ≠ f(x₂)
- Let f be one-to-one and onto function with the domain A and the range B. Then its inverse function \(\boxed{ \ f^{-1} \ }\) has the domain B and the range A such that \(\boxed{ \ f(x) = y \leftrightarrow f^{-1}(y) = x \ }\)
- To find the formula for the inverse of a function, solve the equation y = f(x) for x and interchange x and y
Learn more
- The composite function https://brainly.com/question/1691598
- Let f(x) = x + 8 and g(x) = x2 - 6x - 7. Find f(g(2)) https://brainly.com/question/2142762
- If g(x) = (x+1)/(x-2) and h(x) = 4 – x, what is the value of g(h(-3))? https://brainly.com/question/1900154
Keywords: the inverse of function, f(x) = 2x + 3, one-to-one, onto, domain, range, interchange
