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Here are two functions - OCR - GCSE Maths - Question 13 - 2019 - Paper 5

Question 13

Here are two functions. Function A $x \to x \times 3 \to -2 \to y$ Function B $x \to x + 7 \to y$ (a) Find an algebraic expression for the output of the inverse... show full transcript

Worked Solution & Example Answer:Here are two functions - OCR - GCSE Maths - Question 13 - 2019 - Paper 5

Step 1

Find an algebraic expression for the output of the inverse of function A when the input is $x$.

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Answer

To find the inverse of function A, we start with the original function definition.

Function A is defined as: $ext{function A: } y = 3x - 2$

To find the inverse, we first swap $x$ and $y$ : $x = 3y - 2$

Next, we solve for $y$ :

Add 2 to both sides: $x + 2 = 3y$
Divide by 3: $y = \frac{x + 2}{3}$

Thus, the inverse of function A is: $f^{-1}(x) = \frac{x + 2}{3}$

Step 2

Find the value $x$ when $z = 4x$.

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Answer

In this case, we find the composite function C, which applies function A and then function B:

First, apply function A: $z_1 = 3x - 2$

Next, we apply function B to the output of function A: $z = z_1 + 7 = (3x - 2) + 7$ This simplifies to: $z = 3x + 5$

Now, we set this equal to $4x$ as per the question: $3x + 5 = 4x$

To solve for $x$ :

Subtract $3x$ from both sides: $5 = 4x - 3x$
Therefore: $5 = x$

Thus, the value of $x$ when $z = 4x$ is: $x = 5$

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