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Find Inverse Functions – Practice Questions & Solutions
Question 1
Find the inverse of the linear function \( f \):
\[
f(x) = 3x - 2
\]
View Solution
\[
y = 3x - 2
\]
\[
x = 3y - 2 \implies y = \frac{x + 2}{3}
\]
\[
f^{-1}(x) = \frac{x + 2}{3}
\]
Question 2
Find the inverse of the quadratic function \( f \):
\[
f(x) = -x^2 + 2, \quad x \ge 0
\]
View Solution
\[
y = -x^2 + 2
\]
\[
x^2 = 2 - y \implies x = \sqrt{2 - y}
\]
\[
f^{-1}(x) = \sqrt{2 - x}
\]
Question 3
Find the inverse of the quadratic function \( f \):
\[
f(x) = x^2 - 2x, \quad x \ge 1
\]
View Solution
\[
y = x^2 - 2x
\]
\[
y + 1 = (x - 1)^2 \implies x - 1 = \sqrt{y + 1}
\]
\[
f^{-1}(x) = 1 + \sqrt{x + 1}
\]
Question 4
Find the inverse of the rational function \( f \):
\[
f(x) = \frac{2}{x}
\]
View Solution
\[
y = \frac{2}{x}
\]
\[
x = \frac{2}{y}
\]
\[
f^{-1}(x) = \frac{2}{x}
\]
Question 5
Find the inverse of the rational function \( f \):
\[
f(x) = \frac{x + 1}{x - 1}
\]
View Solution
\[
y = \frac{x + 1}{x - 1}
\]
\[
y(x - 1) = x + 1 \implies yx - y = x + 1
\]
\[
x(y - 1) = y + 1 \implies x = \frac{y + 1}{y - 1}
\]
\[
f^{-1}(x) = \frac{x + 1}{x - 1}
\]
Question 6
Find the inverse of the square root function \( f \):
\[
f(x) = \sqrt{x - 1}
\]
View Solution
\[
y = \sqrt{x - 1}
\]
\[
y^2 = x - 1 \implies x = y^2 + 1
\]
\[
f^{-1}(x) = x^2 + 1, \quad x \ge 0
\]
Question 7
Find the inverse of the cube root function \( f \):
\[
f(x) = (x + 1)^{\frac{1}{3}}
\]
View Solution
\[
y = (x + 1)^{\frac{1}{3}}
\]
\[
y^3 = x + 1 \implies x = y^3 - 1
\]
\[
f^{-1}(x) = x^3 - 1
\]
Question 8
Find the inverse of the logarithmic function \( f \):
\[
f(x) = \ln(x)
\]
View Solution
\[
y = \ln(x)
\]
\[
x = e^y
\]
\[
f^{-1}(x) = e^x
\]
Question 9
Find the inverse of the exponential function \( f \):
\[
f(x) = e^{x - 1} + 3
\]
View Solution
\[
y = e^{x - 1} + 3
\]
\[
y - 3 = e^{x - 1} \implies \ln(y - 3) = x - 1
\]
\[
x = \ln(y - 3) + 1
\]
\[
f^{-1}(x) = \ln(x - 3) + 1
\]
Question 10
Find the inverse of the logarithmic function \( f \):
\[
f(x) = \ln(x + 2) - 3
\]
View Solution
\[
y = \ln(x + 2) - 3
\]
\[
y + 3 = \ln(x + 2) \implies x + 2 = e^{y + 3}
\]
\[
x = e^{y + 3} - 2
\]
\[
f^{-1}(x) = e^{x + 3} - 2
\]
More References on Inverse Functions