Answer :

Sure, let's find the value of [tex]\( f\left( -\frac{3}{4} \right) \)[/tex] for the function [tex]\( f(x) = 12x^4 + 9x^3 - 8x^2 + 19 \)[/tex].

1. Start with the function:
[tex]\[
f(x) = 12x^4 + 9x^3 - 8x^2 + 19
\][/tex]

2. Substitute [tex]\( x = -\frac{3}{4} \)[/tex] into the function:
[tex]\[
f\left( -\frac{3}{4} \right) = 12\left( -\frac{3}{4} \right)^4 + 9\left( -\frac{3}{4} \right)^3 - 8\left( -\frac{3}{4} \right)^2 + 19
\][/tex]

3. Calculate each term separately:
- Evaluate [tex]\(\left( -\frac{3}{4} \right)^4\)[/tex]:
[tex]\[
\left( -\frac{3}{4} \right)^4 = \frac{81}{256}
\][/tex]
- Evaluate [tex]\(\left( -\frac{3}{4} \right)^3\)[/tex]:
[tex]\[
\left( -\frac{3}{4} \right)^3 = -\frac{27}{64}
\][/tex]
- Evaluate [tex]\(\left( -\frac{3}{4} \right)^2\)[/tex]:
[tex]\[
\left( -\frac{3}{4} \right)^2 = \frac{9}{16}
\][/tex]

4. Substitute these values back into the function:
[tex]\[
f\left( -\frac{3}{4} \right) = 12 \left( \frac{81}{256} \right) + 9 \left( -\frac{27}{64} \right) - 8 \left( \frac{9}{16} \right) + 19
\][/tex]

5. Simplify each term:
- For the first term:
[tex]\[
12 \left( \frac{81}{256} \right) = \frac{972}{256} = \frac{243}{64}
\][/tex]
- For the second term:
[tex]\[
9 \left( -\frac{27}{64} \right) = -\frac{243}{64}
\][/tex]
- For the third term:
[tex]\[
-8 \left( \frac{9}{16} \right) = -\frac{72}{16} = -\frac{72}{16} = -\frac{36}{8} = -\frac{18}{4} = -\frac{9}{2} = -4.5
\][/tex]

6. Now combine these simplified terms:
[tex]\[
f\left( -\frac{3}{4} \right) = \frac{243}{64} - \frac{243}{64} - 4.5 + 19
\][/tex]

7. Notice that [tex]\(\frac{243}{64}\)[/tex] and [tex]\(-\frac{243}{64}\)[/tex] cancel each other out.
[tex]\[
f\left( -\frac{3}{4} \right) = -4.5 + 19
\][/tex]

8. Simplify the final expression:
[tex]\[
-4.5 + 19 = 14.5
\][/tex]

So, the value of [tex]\( f\left( -\frac{3}{4} \right) \)[/tex] is [tex]\( 14.5 \)[/tex].

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