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The functions $f$ and $g$ are defined on $\mathbb{R}$, the set of real numbers by $$f(x) = 2x^2 - 4x + 5$$ and $$g(x) = 3 - x.$$ (a) Given $h(x) = f(g(x))$, show that $h(x) = 2x^2 - 8x + 11.$ (b) Express $h(x)$ in the form $p(x + q)^2 + r.$ - Scottish Highers Maths - Question 12 - 2016
Question 12
The functions $f$ and $g$ are defined on $\mathbb{R}$, the set of real numbers by $$f(x) = 2x^2 - 4x + 5$$ and $$g(x) = 3 - x.$$ (a) Given $h(x) = f(g(x))$, show... show full transcript
Worked Solution & Example Answer:The functions $f$ and $g$ are defined on $\mathbb{R}$, the set of real numbers by $$f(x) = 2x^2 - 4x + 5$$ and $$g(x) = 3 - x.$$ (a) Given $h(x) = f(g(x))$, show that $h(x) = 2x^2 - 8x + 11.$ (b) Express $h(x)$ in the form $p(x + q)^2 + r.$ - Scottish Highers Maths - Question 12 - 2016
Step 1
Given $h(x) = f(g(x))$, show that $h(x) = 2x^2 - 8x + 11.$
Answer
To find , we first need to calculate :
Now, substitute into :
Next, we substitute into the function :
Calculating first:
Next, substituting this into gives us:
Now we expand this:
Combining like terms:
This concludes part (a), and we have shown that
Step 2
Express $h(x)$ in the form $p(x + q)^2 + r.$
Answer
We start with the expression for we found earlier:
Next, we factor out the coefficient of the term:
Now, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of (which is -4), square it, and add/subtract it inside the parentheses:
This can be re-arranged as:
Expanding that gives:
Finally, we simplify:
Thus, we express in the form where , , and