Q1: If ax + b is the inverse of the function below, what is the value of a?
f(x) = x + 0.5
Q2: If ax + b is the inverse of the function below, what is the value of b?
f(x) = x + 0.5
Q1: If ax + b is the inverse of the function below, what is the value of a?
f(x) = x + 0.5
so y = x + 0.5
inverse
x = y + 0.5
or
y = 1 x - 0.5 = a x + b
To find the value of a and b for the inverse function, we can follow these steps:
Q1: If ax + b is the inverse of the function below, what is the value of a?
f(x) = x + 0.5
Step 1: Write the inverse function by swapping x and y:
x = y + 0.5
Step 2: Solve the equation for y:
y = x - 0.5
Step 3: Compare the inverse function with the general form of the inverse function ax + b:
ax + b = y
By comparing the coefficients, we can see that a = 1.
Therefore, the value of a is 1.
Q2: If ax + b is the inverse of the function below, what is the value of b?
f(x) = x + 0.5
Step 1: Write the inverse function by swapping x and y:
x = y + 0.5
Step 2: Solve the equation for y:
y = x - 0.5
Step 3: Compare the inverse function with the general form of the inverse function ax + b:
ax + b = y
By comparing the constant terms, we can see that b = -0.5.
Therefore, the value of b is -0.5.
To find the inverse of a function and determine the values of a and b, we need to follow these steps:
Step 1: Start with the function f(x).
Step 2: Replace f(x) with y.
Step 3: Swap the x and y variables. The new equation will be x = y + 0.5.
Step 4: Solve the equation for y. Subtracting 0.5 from both sides, we get y = x - 0.5.
Step 5: Replace y with f^(-1)(x) to represent the inverse function.
Step 6: The inverse function is given as ax + b, so we have the equation f^(-1)(x) = ax + b.
Now, we can answer the questions:
Q1: To find the value of a, we compare the corresponding coefficients of x in the original equation f(x) = x + 0.5 and its inverse f^(-1)(x) = ax + b.
Comparing the coefficient of x in both equations, we can see that a = 1.
Therefore, the value of a is 1.
Q2: To find the value of b, we compare the constant term in the original equation f(x) = x + 0.5 and its inverse f^(-1)(x) = ax + b.
Comparing the constant term in both equations, we can see that b = -0.5.
Therefore, the value of b is -0.5.