Question

If \[f:R \to R\] and \[g:R \to R\] are defined by \[f(x) = 2x + 3\] and \[g(x) = x^ 2 + 7\], then the values of \[x\] for which \[g\{ f(x)\} = 8\] are
A. 0, -6
B. -1, -2
C. 1, -1
D. 0, 2

Answer 1

Hint:We put the given equation \[f(x) = 2x + 3\] in place of x in the equation \[g(x) = x^ 2 + 7\] and simplify it to find the required answer.

Formula Used: The composite function denoted by (g o f) (x) = g (f(x)).

Complete step by step solution:
Here, the given equations are
\[f(x) = 2x + 3\]
\[g(x) = {x^2} + 7\]
\[g\{ f(x)\} = 8\]

Now we put the value of f(x) into g {f(x)}
\[g(2x + 3) = 8\]
Next we put in values of \[f(x) = 2x + 3\] into \[{x^2} + 7\] which will be equal to 8 since \[g\{ f(x)\} = 8\].
\[ \Rightarrow {(2x + 3)^2} + 7 = 8\]
Subtract both by 7
\[ \Rightarrow {(2x + 3)^2} + 7 - 7 = 8 - 7\]
\[ \Rightarrow {(2x + 3)^2} = 1\]
Solving the derived equation further we get,
\[ \Rightarrow 2x + 3 = \pm 1\]
Subtract 3 from both sides:
\[ \Rightarrow 2x = \pm 1 - 3\]
This gives us two values for x
\[x = - 1\] or \[ - 2\]


Option ‘B’ is correct

Additional Information: There are various methods to solve a quadratic equation. The methods are factorizing method, completing square, quadratic formula, using square root method.
 The range of a composite function is the intersection of the range of both functions that are inner function and outer function.

Remember \[\phi\] is a subset of all sets.

Note: students often do mistake to calculate \[g\left( {2x + 3} \right)\]. They put \[x = {x^2} + 7\] in 2x + 3. But the correct way is: we have to put \[x = 2x + 3\] in \[{x^2} + 7\].